With the extra resistor at the output of the integrator I see 3 possible effects:
The first is a nonlinear effect of U206, similar to what I have seen with the OPA172. The bandwidth / phase reserve can depend on the output current.
As 2nd point the output-cross over distortion of U206 would shift to a different point in the curve. Output cross over would usually change the ringing when the output current changes sign. A 3rd effect is that the extra load resistance would give more coupling to the supply, so one would expect more "ripple" on the supply and maybe ground.
For the 3 rd effect one could try some local decoupling for U206/U207, so that AC currents between those two OPs would not flow so much via the global decoupling and through ground. This may need also high capacitance like 100 µF, not just a few 100 nF.
For the first point having the resistor to +15 V instead could in deed help. One could also consider a different OP for U206 (e.g. OPA604, OPA134, TLE2071, OPA140). There where some earlier tests with different OPs, but with not so much visible effect (could be due to not so good tests).
Another relatively simple try could be increasing R220 and thus less current towards the slope amplifier. Even some 5 K would not add that much noise (mainly for short conversions) and should not add too much delay.
@Dietert:
Aliasing between the test points and periodic errors is possible in theory, but unlikely to cause the odd shape. A similar form for the INL error were also found by others.