Band-pass op-amp structure with parallel diodes in feedback?
Does this mean that this is essentially just a bandpass filter that
also clips the output?

Yes, its essentially doing this but with the gain of the opamp. So the output is clipped which is useful if the next stage (or ADC) has a smaller input voltage range.

Source: http://technlab.blogspot.com/2018/07/study-of-diode-as-clipper-clamper.html
The easiest thing to do would be to just simulate that stage in spice (or Lt spice) and do a DC operating point simulation. (I might add one later if I have time).
One way to analyze these is to do a regular gain analysis without the diodes and then draw the point where the diodes begin to turn on, and at that point the output will start to be clipped.