If you like the Rega, get yourself a Rega. If you don’t mind used there are a few P3’s here for good prices now.
Regas are good at what they do - play LPs. A set it and forget it turntable. I have owned a P5, P7 & a P9; all very good at their price points. Regas knock is the adjustability of the TT. No VTA without putting spacers under the arm, no azimuth adjustment for that last little bit of music extraction, and the lightweight plinth. For your budget I would stay within the Rega lineup - user friendly tables. I suggest spednig 1/3 of budget on cartridge and the remainder o n the TT.
Rega has it own sound so if you like it go with that. I own a P7 and I paid just a little over what you want to spend (it was a good deal) but the beauty of the P7 is there is nothing you really need to upgrade other than a spacer if you need them. I don’t fret over a millimeter with VTA. So I put money in the cartridge and kept upping the phono stage over time. Bang for your buck I would consider a used P3 and getting the best cartridge you can within your budget. Oh and spend $50 on a protractor. I use an AVID, set it, forget it, and listen to music.
Azimuth adjustment is often the difference the difference between good and sucks . And easy to do by ear if you can easily move up and down a bit .
On Rega its a PITA . IMO you also need to send a Rega out to have the tone arm rewired before you play a record with those crappy clips they use .
I owned several .
I was in the pretty much same situation a few years ago - went from a Denon to a Rega P3-24 with a Rega Exact, and loved it. I eventually wanted to go up to the next step, and was patient, and bought a highly upgraded RP3 (full Cardas rewire, TTPSU, all the top of the line Groovetracer stuff, etc.) here on Audiogon for a great price. I put a new Exact on it, and I am pretty much set - sounds fantastic, and unless I am willing to upgrade all my other gear, all I think I need. Good luck with whatever you decide.