Where to go next with the turntable

I don’t know turntables at all anymore, but it’s true that adding mid-hi frequency absorption and diffusion will help the bass emerge.

Also, a highly resonant room can benefit from bass traps, further improving the bass so that unnatural accents are gone.

You might want to listen to some digital and see if you feel you have the same bass / balance issues.

Thought I would throw in the Dr Feickert Volare as an upgrade.  I love mine!!!

I bought the P3 and added an RB300 to it and it’s still a cheap particle wood/pressboard plinth with Formica-like skin and a non adjustable VTA -(unless you buy some third party shims). Depending on what non-Rega cartridge you want to pair with it you will possibly have to add the shims to it. I don’t like Regas proprietary approach to their turntables but I’m sure there are many out there who are very satisfied with them. I don’t mean to disparage anybody’s fondness for those products but I regret taking that path and now overlook anything Rega. There are better choices out there for sure.

FYI, 
The room is custom built to be a listening room and there are room treatments. Digital sounds just fine, so no adjustments to the room will be needed. 

Don't get me wrong, the sound is great and the Hegel is a big improvement from what I had, We're talking just a bit of warmth along the level of a cable change (which is also something to consider). 

There are corner traps behind the speakers as well as treatments at the first reflection points as well as diffusers on the wall behind me. Guess I could stand to update my profile :-)

you might want to try changing the cartridge 1st, then later the TT, use the new cartridge on the new TT's tonearm.

The best bass I ever had was a combo of the Thorens TD124’s 9lb platter, SME 3009 tonearm (with removable headshell) with Shure’s last V15vxmr cartridge with beryllium cantilever. Wonderful in all respects, until I broke the brittle cantilever. I now have a new stylus: Jico SAS on boron, in that Shure body. Jico has a brush, but it is not damped like Shure's were.

I have found, the stiffer the cantilever, the better for bass. New, I buy boron cantilevers (not the stiffest, but darn good!). Occasionally I buy low-use used, like a Sumiko Talisman S with a sapphire cantilever that sounds terrific.

I also have VAS re-fit a broken cartridge, the latest is simply un-available without re-building: AT33ptg/II MC MONO Body, new boron cantilever, new advanced tip.

And Shure's 97xe with factory damped brush with new boron/ML stylus for the occasional warped LP.

And, of course I could not do any of this with ease if using a fixed arm.