My thought was that if turning on the backlight was helping, then maybe the voltage is too high and possibly dead batteries might help... but often there's a switching power supply that boosts up the voltage regardless of how low the batteries get. If that's the case it may be strictly the additional load that's helping.

btw, here's something to try... with one of the keys you see the problem with, try programming a macro. Send some key first that'll be ignored. Then send the real key.

Maybe the remote is having issues waking up and sending the first key ...