[mikefocke]

Some thoughts.

First fully charge the battery to its max appropriate level (AGM 12.8+, conventional 12.6) . Load test it. (Battery shops will often do it for free.)

Do this to determine if the battery has gotten tired by being discharged below a certain level many timess. Eliminate that first. Because no matter what else you find, that needs to be right.

OK, now with one of the three items (battery, maintainer, adapter) known good you still would have a problem. I see two choices.

Find a connection point where the power is switched off when the car is turned off. Research first to see what will happen to the settings in the adapter when there is no power. Are the settings maintained. Connect the adapter that way..

Or, get a charger/maintainer. These units will fully charge at a higher rate and then switch into maintainer mode.

Isn't the car still looking for the fob even when it is not near and the car otherwise off? Depending on the model year there is a drain that tapers down after the car is switched off and reached a low after 31 or 61 minutes depending in MY.