Record clamp

There used to be a company called EON Research that made "The Pod".  It was a lightweight, plastic 3-point clamp that worked pretty well on the short Rega spindle.

Then it was available from Ortofon and Pickering and I think a few others.

If you can find one, it might be worth a try?

https://www.vinylengine.com/library/eon-research/pod.shtml

I had a Rega Planar 3 back in the early 90’s.  I tried a few clamps and weights which were available at the time.  Didn’t really make a difference and I never felt comfortable with a clamp/weight on the very short record spindle. I believe there was a quarter inch to three eights inch  length of the exposed spindle for the weight and it scared the crap out of me.  I was using a Sumiko BP Special (the naked cartridge) and was never crazy about the sound of the combination.  I moved on to a VPI HW19 MK III  and used the Rega RB300 arm with the VPI. 

Still the best design, light and won’t cost an arm and a leg.

https://theguruchoice.com/record-players/best-record-clamp/#record-doctor-record-clamp

https://www.amazon.com/Record-Doctor-Clamp-Black/dp/B077YVJW1T

Cheers George

On MC’s recommendation, I’m trying the Origin One Gravity Record Weight on my highly modified Rega Planar RP3 turntable with a Clearaudio Virtuoso cartridge.
Initial testing shows slight but definite sonic improvement.
The overall sound is smoother yet more detailed with some improvement in sound staging. Not a night and day difference, but a nicer listening experience.

I doubt the record weight will harm your table. I cannot comment how if will affect your sound.

I suspect your sound is compromised by the very large quality/performance gap in turntable and cartridge choice. The record clamp will not remedy that. 

I'd advise you to pull the cart and keep it handy for when you upgrade your turntable. Rather than a clamp, invest in a cartridge with a performance more inline with that of the P3. Won't hurt to do both cart and clamp though.