Would increasing the gain in the amp give more weight to the sound? Any negatives?

In integrated amps with passive preamp section it's common not to have full body sound, but transparent. I heard Creek amps with passive section and described them as Brittish-polite and shy amps. I like'm somewhat rude and brave -- in other words completely opposite.
It's not good idea to increase gain of power amp. In most cases the amp can loose stability and become microphonic, pick up emi and simply self-oscillate and hum. 
So theoretically and most trivial, you probably need active preamp and use your integrated as poweramp only.

The integrated has no either pre out or power in, so I would have to use regular line in input, this will bring compatibilty issues. Besides, I see nothing under $2k-$3k new worthy of my attention, plus $500 used interconnect, at least. Putting active preamp, preferably tube, in between phono and integrated is, of course, the first idea that I had.

All you need to do is to move your volume control all the way up and use one of line inputs of your integrated with preamp of your choice. Otherwise to move the right way you simply need to go to separates from your integrated amp.