Consider what happens to the input and output voltages when the supply (either one) is cut to a subcircuit. Almost all ICs have input and output protection diodes. Few have "disconnected when unpowered" pins. More often, "micropower" devices come with enable pins which allow the chip to remain supplied (and thus the input/output voltage range stays the same), while disabling internal bias (resulting in microamperes of current draw).
If your measurements are between seconds, then you must not need much bandwidth. Surely you can find precision micropower op-amps for the task?
Also, you can't disable an active filter. At best, you can freeze it, and it will retain the last state, but more likely it will end up reset to zero. So your settling time is going to be terrible, especially if the bandwidth is anywhere near what your sample rate suggests (i.e., <1Hz).
Overall, this sounds like a project that hasn't had any coherent understanding or attempt at saving power -- if your amps consume "rather a lot", it probably doesn't matter one iota whether the CPU is running balls-out or not, does it?
So, simplify the problem, make an analogy: something easier to work with. What if you were doing it completely low level with state-of-the-art '70s tech? You could have:
- Analog input stages
- Power/bias enables, analog switches, S&H, etc. as needed
- ADC (of whatever type -- D-S, flash, SAR..), parallel out
- SRAM -- CMOS, with battery backup (microamperes -- only draws significant current during pin / state changes)
- Digital logic -- address counter, clock and divider, etc. (also CMOS)
- Maybe a bus arbitration thing to allow outside access (e.g. reading it out via parallel port) from a powered device, as well as other assorted logic for housekeeping (address reset, buffer overflow stop, battery "low" indication or logging, etc.)
With a little more work, a serial ADC (much more common these days) and serial EEPROM could be used, plus whatever interface method (e.g. RS232 ASCII terminal?); at this point, you'd be looking at a low power FPGA to handle all the sequential logic, or even some software.
It's pretty clear that, however you cut the digital stuff (all in MSI hardware, or embedded, or bit-banging it all in software), it's not a threat, at all.
If you can better define the analog requirements (voltage or current, and how much of each, precision, bandwidth, sample rate, noise, etc.), we can offer better comments / example scenarios.
Tim