Here's something that might or might not be relevant.
I bought a used Subaru Outback two years ago and the fob (actually both fobs) acted strangely from the day of the sale. The car dealer just said 'those fobs behave differently'. Total BS as my previous Outback fob worked normally. I went to the Subaru service place and the guy at the counter said, 'Your fob has a broken battery contact, happens all the time. It will be $30 to fix.' I asked him how he was so sure without seeing it. He asked for the fob, got a little screwdriver from a pen jar on his desk, took the back off and showed me the contact. 'See that? It breaks away from the circuit board and causes intermittent problems. We get one a week like this. Mostly it's because parents give the keys to kids to play with and they get dropped.' My car had been the Mum's kid-ferry so that made sense. "We just solder the connector back onto the circuit board better than it was when new, total fix.' I had him fix both fobs, no problem after that. It's sound a bit far-fetched that both your fobs have a similar problem, but perhaps it's worth seeing if the battery contacts are loose inside.
From Woodrising (no, nobody else has heard of it either) Rides a Springfield Dark Horse