Chopper Amplifier feedback loop stability
Yes, stability will likely be a problem, and the internal construction has little to do with it. Most (modern) chopper amps have many MHz bandwidth and behave similarly to normal op-amps or in-amps aside from really nasty spikes out the inputs and some small noise near the modulation frequency.

However, you are introducing a lag and more gain into the feedback loop and both of those will tend to result in less phase margin and thus potential instability. By keeping the gain of the in-amp low and perhaps introducing some compensation you should be able to make this concept work.