You need to check the voltage noise of the regulator outputs with the opamps connected. Probably you did it. If so, it is a riddle why the 8kHz bump is not observed in the regulator outputs, but is in the opamp outputs...

Does the noise bump happen with the +/-15V supplied by 7815/7915? How about to change the capacitor values for LM317/337 to the ones recommended in the data sheet?

It is great to see the noise peaks were largely reduced by LT1792. This is what I found before although I can't explain why.