All of my basses are tuned in fifths. I have two four string long scales, two six string short scales (but I converted one to a five string), and a Squier 51 which is a Stratocaster that is styled to vaguely resemble a 51 P-bass and which I converted to a four string for now. I may go back to six strings on that one but that is another story! The long scales are not difficult to find suitable fifths strings for at either Bass Strings Online or Kalium.
My six string short scale is a Squier VI, basically a long scale, three pickup Jazzmaster guitar that is intended to be tuned like a guitar but an octave down. Labella makes strings for it and allow you to order custom sets where you pick the gauges for each of the six string from their fairly broad selection of gauges. I tune the bottom string on that down to C using a 120 gauge string. Other brands make strings for the VI too but I cannot tell you if they allow you to order custom sets. I think that string availability for this particular instrument is aided by the fact that it appeals to both bassists and to guitarists who want a deep baritone instrument and both camps use a variety of tunings. If you want to get one they are still available new from Squier and they are in your price range. A lot of bassists shun them because they resemble guitars too closely and they have a guitar's very narrow string spacing. Someone coming from banjo might not have those hangups and in fact some of us bassists don't have them either! I played mine in the church band yesterday with a lineup that was reduced to me, synth, and piano due to everyone's summer vacations. I am a really lousy excuse for a guitarist but that instrument tuned in fifths has a HUGE range and I could easily use my limited skills to fill in for the missing guitars in a couple of places. Again, a banjo player can undoubtedly make much better use of that range than I. My CGDAEB tuning is the original vision that Robert Fripp had for his New Standard Tuning (NST) for guitars but an octave down. I read that it is difficult to impossible to get a reliable B string when you tune an octave up on a 25.5 inch scale but that has not been an issue so far when tuned down an octave on a 30 inch scale.
Sorry for the extended infomercial on fifths tuned Squier VIs but it does seem like an option that you should consider and it is easy to overlook. I don't play mine that much but as I rehearsed and then performed with it last week I realized that mine really deserves more love from me!
My other short scale is an Ibanez SRC6 that I converted to a five string, an SRC5 if you will. The wider spacing you get on that with only five strings feels more natural to me but you might be happy with one just as they come from the factory. New I think they are outside your price range but I got mine used from Guitar Center for about $400. There is a classifieds section here on TalkBass where you can probably find a used one from time to time as well.
For the Ibanez and the Squier 51 I cut down long scale bass strings. "Everyone" here on TB says you cannot do this because the strings will unravel but if you listen carefully there are a few of us here who claim to do it successfully. My approach is to fit the string to the bass, mark where I want the outer windings to end, and then put something on the string between the nut and the tuning post to keep the string from unwinding. Originally my plan was to crimp some short pieces of brass tubing that you can get from a hardware store on the string. But when I actually went to do it one night I just decided to use some electrical shrink wrap tubing that I had in the basement instead of brass tubing that I would have to go out and buy the next day. That seems to work fine for round wounds and pressure wound strings. I don't know if it would be successful with flats.
The Squier 51 is available only used now as far as I know and converting one to a bass may be more of an adventure than you are up for. Even if you leave it at six strings I don't know if the existing bridges on guitars will intonate properly with heavier bass strings. I don't tune the 51 as low as the others which all start on C. My 51 is tuned GDAE right now although I have given some thought to tuning FCGD to get a little closer to the traditional bass low note. There is a thread here on TB called Sub Short Scale Bass Club or something like that where guitar to bass conversions are discussed. If you have an interest in that you can check that out.
The Ibanez Mikro and the Squier Short Scale Jaguar are both available in your price range and both have a decent reputation among owners here. The Jag is a four, the Mikro can be had with four or five strings. If you are ok with a six string guitar tuned down to the bass range there are 27 to 29 inch baritone guitars that you can find in your price range used for sure and maybe there are some in your range new too. Squier also makes a 30 inch hard tail Jazzmaster, a few years ago Fender made a nice Telecaster baritone you might find used somewhere, Eastwood makes a few, and I am sure there are others but I am not familiar with.
Sorry for the novel length post but I think you can find something suitable out there and you should be able to get strings for it from Kalium if nowhere else.
My six string short scale is a Squier VI, basically a long scale, three pickup Jazzmaster guitar that is intended to be tuned like a guitar but an octave down. Labella makes strings for it and allow you to order custom sets where you pick the gauges for each of the six string from their fairly broad selection of gauges. I tune the bottom string on that down to C using a 120 gauge string. Other brands make strings for the VI too but I cannot tell you if they allow you to order custom sets. I think that string availability for this particular instrument is aided by the fact that it appeals to both bassists and to guitarists who want a deep baritone instrument and both camps use a variety of tunings. If you want to get one they are still available new from Squier and they are in your price range. A lot of bassists shun them because they resemble guitars too closely and they have a guitar's very narrow string spacing. Someone coming from banjo might not have those hangups and in fact some of us bassists don't have them either! I played mine in the church band yesterday with a lineup that was reduced to me, synth, and piano due to everyone's summer vacations. I am a really lousy excuse for a guitarist but that instrument tuned in fifths has a HUGE range and I could easily use my limited skills to fill in for the missing guitars in a couple of places. Again, a banjo player can undoubtedly make much better use of that range than I. My CGDAEB tuning is the original vision that Robert Fripp had for his New Standard Tuning (NST) for guitars but an octave down. I read that it is difficult to impossible to get a reliable B string when you tune an octave up on a 25.5 inch scale but that has not been an issue so far when tuned down an octave on a 30 inch scale.
Sorry for the extended infomercial on fifths tuned Squier VIs but it does seem like an option that you should consider and it is easy to overlook. I don't play mine that much but as I rehearsed and then performed with it last week I realized that mine really deserves more love from me!
My other short scale is an Ibanez SRC6 that I converted to a five string, an SRC5 if you will. The wider spacing you get on that with only five strings feels more natural to me but you might be happy with one just as they come from the factory. New I think they are outside your price range but I got mine used from Guitar Center for about $400. There is a classifieds section here on TalkBass where you can probably find a used one from time to time as well.
For the Ibanez and the Squier 51 I cut down long scale bass strings. "Everyone" here on TB says you cannot do this because the strings will unravel but if you listen carefully there are a few of us here who claim to do it successfully. My approach is to fit the string to the bass, mark where I want the outer windings to end, and then put something on the string between the nut and the tuning post to keep the string from unwinding. Originally my plan was to crimp some short pieces of brass tubing that you can get from a hardware store on the string. But when I actually went to do it one night I just decided to use some electrical shrink wrap tubing that I had in the basement instead of brass tubing that I would have to go out and buy the next day. That seems to work fine for round wounds and pressure wound strings. I don't know if it would be successful with flats.
The Squier 51 is available only used now as far as I know and converting one to a bass may be more of an adventure than you are up for. Even if you leave it at six strings I don't know if the existing bridges on guitars will intonate properly with heavier bass strings. I don't tune the 51 as low as the others which all start on C. My 51 is tuned GDAE right now although I have given some thought to tuning FCGD to get a little closer to the traditional bass low note. There is a thread here on TB called Sub Short Scale Bass Club or something like that where guitar to bass conversions are discussed. If you have an interest in that you can check that out.
The Ibanez Mikro and the Squier Short Scale Jaguar are both available in your price range and both have a decent reputation among owners here. The Jag is a four, the Mikro can be had with four or five strings. If you are ok with a six string guitar tuned down to the bass range there are 27 to 29 inch baritone guitars that you can find in your price range used for sure and maybe there are some in your range new too. Squier also makes a 30 inch hard tail Jazzmaster, a few years ago Fender made a nice Telecaster baritone you might find used somewhere, Eastwood makes a few, and I am sure there are others but I am not familiar with.
Sorry for the novel length post but I think you can find something suitable out there and you should be able to get strings for it from Kalium if nowhere else.

