rickman wrote in :

For an opamp to work as an opamp it needs to have
significant gain over the BW in use.

Ok, how about just enough gsain to get a buffered output of some oomph to
survive integration to slow clean pulses? That might not take so much to do,
and if it works, it really takes the strain off the real gain stage
which follows it because that will be operating pretty much at DC capability.