Pro Co Rat Bass mods as my first undertaking...

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Looks great!  There's something really cool about those trimmers sitting up high like that.

Audio pots will only help if you're finding that most/nearly all of the change happens in the last 10% or so of the pot rotation.  I don't think you can get trimmers in audio taper though, but I could be wrong.  You could try a much simpler approach - replace the 560 resistor with 330 or 270, or replace the 4.7uF cap with 2.7 or 2.2.

I've never done an a/b with and without r16 so the only thing I know about it is what I've read.  R16 is only found in the Turbo RAT, and what it does is to give the output transistor some breathing room so it isn't pushed as hard by the higher output of the Turbo RAT due to the higher clipping threshold of the LEDs.  It's normally 2.2M and part of that same Turbo RAT adjustment is to make R10 2.2M as well.  I've been told it's a very subtle difference and kind of sounds like there's more headroom and less compression after the distortion.

There is a little room on the RAT for more stuff.  There's a cavity above the the jacks north of the PCB.  Small switches will fit on the back and top of the RAT above the knobs without too much trouble.  Trim pots will fit there too but the trouble is how to mount them so they are accessible.

Edit: I've just been doing a little math on this circuit and I think the best thing to do if all you want is a little more bass is to change the 560 to 470 or 390.  The lower the value, the more gain you get at the lower frequencies, but at the same time you cut out more and more of the bottom end.  So you don't want to go too low.  Once you go below about 390 you start filtering out low end above the low E string on the guitar, so although gain increases, it is applied less to the lowest frequencies, which will result in louder mid-bass but not low bass.  All of this happens within a fairly small range of values, which is probably why the 1K trimpots aren't working as well as was hoped.  If you really want a trimpot, try a 330 resistor in series with a 250 trimpot - that should keep the entire range of the trimpot fairly useful.  This makes sense on paper at least - YMMV.