Hey @Panda !
I’ll start by saying that unfortunately this is not going to be an exact solution, rather just an idea for your question.
I don’t know if that was implied, but I don’t even think that you need to “process” the signal in that case, rather what you would need is simple a sort of Bluetooth signal repeater.
On second thought, maybe a bit of processing would be needed in terms of identifying the fob’s signal so that it does not repeat every signal that comes by.
Nonetheless, what I’m thinking about is something that is like the WiFi repeaters that are all over the place basically. Have you checked out if there is some sort of solution in that direction?