Thanks for posting the schematic!
It is an interesting problem. I am not super familiar with low-offset stuff or active offset nulling and so on, so my thoughts were more along the lines of avoiding the issue by applying engineering principles, forgive if I am stating the obvious as I feel it useful to eliminate obvious things first.
You said the input stage is powered separately and from a 5V reference. Have you checked the specs carefully? The current drive it can provide and the capacitive load it can drive without oscillation? It seems risky to me to use a reference as a power supply. Can you swap it out for something like a 78L05 or LP2950? Op amps have very good CMRR so I do not see the need for a fancy power supply. Having more current drive will probably improve regulation more than basing it off a super accurate reference.
Now about the input offset. In principle this occurs because the input current, while theoretically zero, isn't zero. It may be microamps. So even if you hold the input at PRECISELY some reference voltage, the voltage the op-amp sees at the input will vary slightly due to the input current and the tiny voltages that this develops e.g. if there is input resistance.
To a large extent this is unavoidable and it also should not matter overly much since it mostly just behaves like an extra input resistor (or impedance) or an extra feedback resistor or an extra resistor to ground or similar, and this should have fairly repeatable and predictable effects through the output swing and the input operating range. Input current can be affected by temperature, but you say the temperature is not changing significantly. So it does seem mysterious so far.
The above analysis assumes you are holding the inputs at precisely the wanted voltages and any offset error is internal to the op-amp. I suspect this is not the case. My concern is that changes in the input offset current is changing the voltages in the resistive divider networks feeding the inputs, because of the high value resistors you have chosen.
With those 1M resistors you'll get 1 uA per volt so about 5 uA. An input offset current of just .1uA could change the current in the upper or lower leg of the divider by as much as 2%, you're not seeing such an extreme drift however I think you can see my point that such large resistor values are risky. I could see why to use 1M dividers in battery powered equipment but that's normally in places where some drift or offset is unnoticeable, and I would never use 1M in mains powered equipment at all, I would aim for 100k usually.
Theoretically, if you have only negative feedback and all factors such as temperature, supply voltage and RFI are controlled, then it is totally impossible for your output to drift over time. However, it can happen if there is some positive feedback in the system. And I think that is your case. As you use different values at the different inputs (680k, 1M etc) then a tiny fluctuation in offset current can have a larger effect at the positive terminal than negative hence causing the drift to become self reinforcing until another fluctuation.
You said the filter can't be changed. So you can't change the 680k to 68k? Then what about buffering it with a high quality op-amp... NE5534? Anyway. I must say that blanket statements about what can and can't be changed are not super helpful. Of course a minimal change solution may be sought, but surely if the problem turns out to lie with a particular part then it would be sensible to change it.
Anyway, the things proposed here are easy to test. I would have thought that once you have the test operational it would be a fairly simple matter to swap things until you find the source of the drift. Hopefully I gave some ideas to try.
cheers, Nick