Most locksmiths can test the fob in a couple of seconds.
If the fob is OK and the problem is intermittent and is a problem unlocking, it is often a wiring problem, if its a problem locking its worth checking that the ignition barrel is not sticking in when you take the key out, this is often seen as a problem locking that works OK when you wait a few seconds before locking while the barrel pops-out.
You can tell if the signal is getting to the BeCM in seconds with a meter, check the voltage on the orange wire (pin 6) on the blue plug at the front of the BeCM, it should show 2v normally and 10v when the remote button is pressed. This will help you find out if the problem is before, or within/after the BeCM
If you dont get 10v at the BeCM, check at the receiver under the drivers side rear shelf, single plug with one wire at the receiver should also show 10v when the button is pressed.
The rear doors work on a different circuit to the front doors so that explains why their operation is different.