XLR to RCA Adapter Dilemma

... the sound is much fuller than when going from RCA to RCA. Could this be because the XLR output has a higher gain than the RCA?

Yes, that is normal. The sound quality has nothing to do with RCA or XLR Design, when you have both it depends on matching components and what technical quality the Designer has implemented.
RCA/XLR adaptors degrade the sound normally because they are internally super lousy wired. Open them and you will find even in very expensive adaptors super thin, ultra cheap wire. When you want a good one, contact SignalCable, they do custom order with Silver wire, not expensive and very, very good. You can also order better connectors (for signal transfer).

Unfortunately, my amp is not at all balanced, and I'm not looking to replace it since I like it just fine, and would rather save the dough. So, for now, it's adapter city for me. I may just contact Signal Cable and inquire about custom adapters. Thanks.

A lot of amps have an opamp on the input to handle the XLR input signals, then come back to the single end after this opamp. Where the rca input bypasses this opamp.

I have found this on some big hiend amps. You can imagine that the rca on these amps sound better than the xlr inputs when the levels are the same, but being an opamp for the xlr input some have a small amount of extra gain especially if the opamp they used is not unity gain stable.
These kind pseudo xlr inputs are more common than you think.

Cheers George

Hi - check out the Sterophile review of the BAT 3iX. Although this pre is different, the balanced output has 6db higher gain when used with single ended input.
Furthermore, the output impedance of the preamp varies significantly with balanced/unbalanced across different frequencies and in the testing there was a significant hf roll off in balanced mode with low impedance input power amps. Your power amp is 31k input impedance and the VK3iX is high and goes to 14k at some frequencies.
If the 3i is similar to the 3iX then you probably have a hf rolloff issue in balanced mode with the Musical Fidelity amp.