I just searched for the old thread about turning fobs off and at least one user said his 2013 fob could not be turned off. Here’s that thread for anyone interested:
Turning off key fobs
I agree working through the ideas mentioned above and hope they solve the drain problem for you.
As a secondary issue, Is your 2010 fob like my 2020 fob where you can use a key sequence to put the fob to sleep? Turning off that signal broadcast would be useful for saving battery life, but...