Upgrading Turntable suggestions

Oh no, I am not sending my turntable to California for Brian to dial it in, and he doesn't sell Nottingham.

The Rega RP10 with Apheta cartridge is supposed to be superb and squeezes into your budget

You will be disappointed with just replacing your turntable.  Sure there will be some  improvement in speed stability, etc., but I'd look elsewhere for improvement that is worthwhile.

I recently felt the need for something to spark my moribund sounding vinyl.  With discussion/reading, I settled on a VPI Prime TT with Ortofon black cart and Lehman Black Cube Original pre amp.  HUGE improvement and I am back into analog again!  My original TT was a Project Debut Carbon.  I have ML ESLs, soon to step up to EXLXs.  Bought from SoundStateDirect, online.  Satisfied with them so far.        

Why not try a proper vintage Direct Drive turntable instead of that overpriced modern belt drives ? I’m pretty sure that Denon DP-75, Victor TT-81, Technics SP10mk2 and Technics SP-20 are superior to any belt drives and they are not expensive compared to modern turntables. Powerfull motors, immediate start/stop, speed stability and amazing build quality - this is all about Direct Drive turntables, not about belt drives. It’s better to spend more on plinth, tonearm and cartridge.

I have those Technics SP-20 / SP-10mk2 and Luxman PD-444 in use
The PD-444 is the most expensive of them (but still within your budget), but the lower priced version PD-441 is also very good (it was made by Micro Seiki) if you're looking for something to plug and play. The armbases of Luxman is fine for almost any tonearms. In this sutuation spending $2500 on belt drive is a waste of money.