Buying a new TT today

this one might be beyond your reach, but messing around in the price range you are in will likely eventually result in a similar net expenditure.

the Rega P10 ’package’ at $6000 retail will boost every piece of the current package you have to 2 or 3 levels above. and it’s a coherent, synergistic package.

https://www.stereophile.com/content/rega-planar-10-turntable-rb3000-tonearm-apheta-3-phono-cartridge

and if you can sell what you got, and get a 'deal' on this one......there you are.

Until things settle down and you can actually go out and audition different deck's I would suggest simply updating the cartridge and/or stylus.

For example you could replace your Red stylus with a Blue stylus for well under $200.

For not a whole lot more you could try the Bronze version as these Ortofon's would seem to be a good match for your tonearm (considering that the manufacturer supplies it with the deck).

In these times I would simply go for the Blue stylus, which is said to be a nice improvement (more weight to the sound, et cetera).

Then you would end up (eventually) with a decent backup cartridge with Two usable styli.

DeKay

What Mike says.

Thing about cartridges is that they wear out. And they wear out faster on cheaper equipment. And don't sound as good on cheaper equipment, or on incompatible equipment. So, what Mike says.

I think if you look long term, you should look at your whole system and try to figure out how you can get the most improvement for the $$.

if you are dead set on investing $2K on the analogue rig, Rega tables offer the best value at the low/mid end and hold their value incredibly well. They offer discounts with their own cartridges, which are OK at their lower end, better at the higher end. When you want to upgrade you’ll get a lot of this investment back. Their arms are the best value of all, and maybe you can get a dealer to put an upgraded arm on it and hold onto that long term and upgrade the cartridge next. Maybe a P3 with a power supply unit (If that’s fits with it) with an RB 880 arm? Otherwise a P6. Talk to a Rega dealer and push for a 10% discount. 
If you’re lucky, you can get a demo or used combo. All I can say is that the tone arm, cartridge and phono stage are the most important, assuming you have a TT that has a consistently accurate speed with good isolation.

Good luck and enjoy going down the rabbit hole and remember, the salesman will always try to get you to spend more as there is always something better. My guy turned an arm wire upgrade into an arm, then the whole table. Previously, my cartridge was the best part of my system,  and now I think it is well matched. That’s what you want.

I just bought a new VPI Super Prime Scout with some nice upgrades included from Upscale Audio in L.A.  I couldn't be happier and it would be a worthy upgrade from your Project.  You should check them out for sure.