An extra buffer between U206 and U207 is odd. It would add some delay that could become a problem during rundown and the zero phase. The current from U206 driving the input to U207 should be covered by local decoupling around U206/U207. So I don't think it should be such a problem.
As one difference between positive and negative input voltage is seen in the length of the comparator pulse (much longer with the positive input), one could test reducing R205 to reduce the hysteresis of the comparator a little. This would shorten the pulses a little and might be enough to avoid U207 to reach it's positive limit, as it is done now with positive input. It's kind of a long shot, but only a small change.
I would consider to also replace U206 with an OPA140 or another slightly faster OP than the LT1056 (e.g. TLE2081, OPA134, maybe even OPA604). As there was quite some effect of loading U206, the output stage of U206 could have an influence. So a good quality socket for U205 and U206 might be an idea.
With just U205 replaced with the faster OPA140 one might have to reduce R217 to something like 220-470 Ohms. The divider after U205 kind of reduces the BW of the extra OP in the integrator. With both U205 and U206 replaced with OPA140 the original 1 K could be about OK. In principle could might be able to tweak the response of the integrator with R217, but the original relatively slow response seems to be good enough.
For the amplifier part U104 noise and drift are not that critical. The JFET input stage has a gain of around 30-80. So drift and noise of U104 would be attenuated by this factor - a little less at higher frequenices. From this the TL071 should be good enough. The bigger problem with the TL071 is the ringing / tendency to oscillate - this might be coped with with changes to the compensation. Another possible factor could be the open loop gain, that is much lower with the TL071, this could effect the linearity of the amplifier a little. However some of the higher gain of the OP07 is at very low frequencies and thus likely not in effect with the AZ mode. Still the TL071 was just a first guess / standard part. Ideally I would like to have something like 2-3 MHz GBW with good phase reserve and high loop gain.
OPA130: rather slow (1 MHz GBW), but still faster than the OP07 and higher slew rate and good open loop gain.
TL031: only slightly faster than OPA130, but less loop gain and higher noise / drift (still not too bad)
AD820: moderate speed (1.8 MHz) and higher loop gain than TL071 so possibly a good match.
LT1055: slightly faster than TL071, but seems to be good phase reserve.
OPA145 (slow brother to OPA140): slightly faster than TL071, but looks like good phase reserve (at least the higher frequency end) and good loop gain. Those faster OPs would likely need a change in compensation. The OPA140 is even faster and would definitely need a different compensation, maybe even with the LT1220 part.
The change in compensation would be slightly lower resistors for R122, R123, R138. For the fast ones with high loop gain (e.g. OPA145) a slightly higher R125 would also be an option. In principle slightly smaller caps C120 / C113 could be possible with a faster OP and changed resistors - this would be for faster response, not for less ringing.