SL/R230: Odd issues with 2005 SL65 AMG

Here are the issues with SL:
- Car does not respond to a remote. When I use the key, open the door, start the car, let it run for a few minutes, shut it down. Car responds to a remote as it should (I can lock/unlock doors).
After 10 minutes, car does not respond to a remote
- I used a battery tester and got nowhere. Same battery tester works well with S65 (which uses the same starter battery). Since SL starts without issues, this must indicate that starter battery is ok (right?) but it makes no sense that I battery tester does not even power on when I connect it to the starter battery!
Starter battery is from 3/2021.
I also noticed that the 12V socket in the car does not work.
Yesterday, when I stopped at the red light, car shut itself off! I had to put in in park and start it again - it started immediately.
Can anyone recommend a good indie Mercedes mechanic in Saint Petersburg, Florida?
Thanks






