Finding a short scale bass wasn't terribly challenging but are there any four string basses with narrower than the typical .75"/19mm spacing? Please keep it to options under $1K.
some bridges will allow you to narrow your spacing considerably...and a bridge would be less expensive than an instrument. just sayin'.
Or, most Hofner clones (much less than $1K); most early '70s Japanese basses - like this one; or Mosrite basses (won't be under $1K, though); or the Mosrite copies....
For short scales, check out the Chowny SWB-1, Guild Starfire and M-85 (they show up used under $1K), Danelectro Longhorn, Epiphone Viola and Rumblekat, and most anything Hofner (the Contemporary line is excellent). The medium scale Aria Pro II CSB/TSB models from the early 80s also have narrow spacing at the bridge and are very good instruments at affordable (used) prices.
Check out different scale basses, short or medium. There'll be a lot more options in spacing with smaller scales than without. The gibson, that someone mentioned earlier, does have smaller string spacing. I didn't care for the feel of the bass or the sound when I tried one though. But, I dont care for the guitar version either. Plenty of people out there do. But, it did look good for sure. That's what made me want to pick it up and play.
I used to own a Vox double cutaway hollowbody that had a neck that was so narrow I couldn't cope with it. I have large-ish hands and it was not easy to play without gutter balling the E or G strings right off the fretboard. I was younger and maybe more ham-fisted, but I had to sell it. Too narrow.
I believe that Soundgear and MM Sterling are both 19mm at the bridge. At least every one I've seen is. They're narrow at the nut (38mm/1.5") but widen to 19mm space at the bridge.
Vintage Aria pro IIs have 16mm spacing. The SB line, TSB line and CSB line. The latter two are also medium scale.
The Squier Bronco bass is under 19mm spacing. I think the spacing is 17 or 17.5mm, and it's way under $1,000. I replaced the original 2 saddle bridge with a 4 saddle one that has 18mm spacing: https://www.amazon.com/dp/B00F0CEF0Y?tag=viglink12772-20
I have a Steinberger XL2 that is very well known for having very tight string spacing. However if your budget is 1k it wont be an option. Not sure about the Steinberger Synapse 4 or Radius basses but it may be worth a check if you can get past the boat oar shape..