In that price range you will have a lot of great options.... My recommendations may be (probably are) already addressed but here they are in no particular order..
1. G&L LB-100- Positives: Made in the USA... choice of neck profiles Many appointments you won't get with other companies (6-bolt neck, awesome G&L bridge, better ergonomics, huge choice of colors)... the bad side is if you get a lot of options you will max out your budget which I don't feel is really necessary if you're just wanting a P bass for some sessions and as a backup instead of using it as your main player... also you will be waiting a little bit, so if you need a Precision immediately this may not be a viable option... decent but heavy case included
2. Carvin P bass- Positives: Made in the USA, graphite reinforced neck, also multi-laminate necks available Your choice of colors and wood as well as fingerboard radius (which could be a huge deal if you want super low action) solid Hipshot hardware available in a 5 string (not something I care about but a lot of people do).... the potential negatives mirror that of the G&L... if you go nuts with options you could spend unnecessary money on a piece you dont plan to use much and you will have to wait while it is built... no case included (you have to buy one for extra when you complete your order unless their policy has changed recently)
3. Fender USA Std - Made in the USA Positives: graphite reinforced neck with the narrower P nut width and taper,great for the classic P sound,good pickup, good price point, easy to find accessories, immediate availability, available in a 5 string...Negatives: varying quality (I wouldn't buy one you can't try out first. Limited color options (not really a big deal)... Fender customer service is pretty inconsistent... includes very nice hard case
4. Lakland Skyline 44-64- Made overseas (Korea I think?) Positives: good price point (well below budget), addictive neck profile, option of a jazz neck (on the 44-64 Custom), really comfortable neck regardless of dimensions, Lakland has incredible customer service and quality control is second to none with immediate availability, 5 string available (5 string only has graphite reinforced neck)... Negatives- no reinforcement on 4 string neck (i dont see this as a big deal... none of those 60s P basses that everyone clamors for have reinforced necks)... hard to get some accessories such as pickguards... extremely limited color options wuth the standard 44-64 but the 44-64 Custom comes in some pretty cool LTD finishes no case included
5. Reverend Fellowship Bass- Completely different from any other P style bass aesthetically... made overseas... Positives- VERY good quality control and customer service. Well below budget.. 12 in neck radius (slam that action!) multi-laminate neck that is very stable, similar bridge to the G&L, set neck (not sure if this is a positive but it is an option you wont find in any other production level P).... Negatives: Neck dive (Reverend is notorious for it... it isnt too bad and can be fixed with Hipshot 1/2 inch tuners for about $100) .... case included
If you want to go full custom, I have a great luthier that will build you a bass that is Fender Masterbuilt quality (I know that's big talk, but believe me) for well under your price point but the negative is that since he does EVERYTHING by hand, you will wait about 3-6 months depending on his workload.
Out of everything, I honestly think that if your going to the store to play a bunch and see what you like the Fender USA std is the way to go... If you;re in a remote area and have to order online, I would go for a Lakland Skyline hands down.... just because you have $1600 available doesnt mean you need to spend it all... both of these options are great for your purposes and come in way below $1600