The module should go to sleep, and the battery management should also load-shed it if it detects the battery draining. I suspect this is probably not implemented correctly on some cars (because what is), but it's definitely something that's tested for regularly (since cars can be expected to be taken camping, off the grid, or even just purchased by owners who live outside of cellular coverage).

> The module should go to sleep

lol tell Subaru owners about this. There are tons of them complaining of batteries going dead the last few years, just from sitting a few days in the driveway, while the always-on cellular is at edge of range, hunting. Subaru's solution to this has been to in some cases pay for a bigger battery for those customers.

Maybe an other way around solution? Is is possible to build some kind of dummy cell tower that supersedes real ones?

Good luck with that. This is illegal (in europe)