OK, so the lamps shine bright when fed 12V directly.
Is the lamp actually feeding 12v in the first place?
There is a transformer in the base, I believe. If you can access that, you should be able to check if it is supplying 12v (probably AC).
Once you have eliminated the transformer it should be a simple matter to check the circuit from transfer to switch to lamp and back again.
If it is a high resistance joint and you are running a halogen bulb, it may be that the troublesome joint may run slightly warm with the lamp set "on" and at high power.