David,
Thank you for your helpful responses. I'm quite exhausted by the time I get home from work, so it often takes a while to get technical concepts to click in my head.
I really want to avoid complicating what is otherwise a fairly minimal circuit to null out offsets. I did consider breaking out those pins on op amps that have them, but on their own they wouldn't do any good if the rest of the circuit requirements weren't met. I might just accept these errors, as they probably wouldn't pose an issue for my needs. At this point it's hard for me to say, but I can go the easy route, play with biasing a bit and if I see big differences, I'll try to design a more precise circuit. This thought also extends to the effects to deviations from target bias voltage due to input currents. It may not be worth the effort to design a more sophisticated multi-stage amp, but time an experimentation will tell. If I have some data rather than just some hunch that more gain will help, then I'll address it when the time comes.
I did come across an interesting circuit in the LTC6244 datasheet in which the current source is fed into the non-inverting input of the first amp with a gain set to 3 and the output of that went to the inverting input of a 2nd amp whose feedback path looped back over to the first amp's input. The point of that was specifically to reduce input capacitance somehow, but I doubt it's just limited to that.
Lastly, trying to wrap my head around the effects of increasing gain some more, if I'm not mistaking, for a voltmeter with a given resolution (which is used as the measurement device), for any increase in open loop gain of the amp, while allowing smaller input deviations for a given output, has the effect of reducing the range of measurable input currents, correct? Hopefully that sentence makes sense.