Low-power Linux-based ARM system on module for data logger?
Regarding the latter, how can the CPU enter sleep mode and keep the OS running? I am open to any suggestions.


You use co-processors.

Phones are doing this for years now. They have one or more big ARM Cortex-A cores for the *unix based operating system. And one or more smaller ARM Cortex-M like processors for the background stuff.
Like counting your steps using the accelerometer or keeping track of where you are by GPS.

The smaller ones may run an entirely separate operating environment and may not even be on the same chip.

You can do that as well, for example with the a system on chip eg: SP7021 with 4xCortex-A7's, one ARM926 and a 8051 core.