Well I'm going to end up buying a new remote just to see if it works better.
The suggestion above though to hard-wire the thing to my car (even though it was a joke) interested me.
Took the PCB out again and got a closer look. I'm going to try hard wiring it and installing it somewhere in my door panel (probably next to the power window and door locks / switches).
I'm going to use this momentary switch:
http://www.performance-pcs.com...info&products_id=23263
The power door locks / switches in my car are black and have orange / amber lighting so this momentary switch would match. I found the spots I could solder this switch to on the PCB.
Then I'll use this to convert my car's 12V line to 3V:
http://www.dimensionengineering.com/AnyVoltMicro.htm
I also found the spots on the PCB I could solder this to (so I can dump the 3V lithium battery). I have a question about connecting everything though. The LED in the momentary switch is 3V. The garage door opener remote is also 3V. Can I simply connect a 12V wire from my car to the Vin of the converter, set the converter to 3V then connect BOTH the LED in the switch AND the garage door opener remote to the Vout of the converter?