There are a couple of pawls inside which can stick and lock the gearchange but the only likely option is to take it apart. Even if it frees off with heat you know there will be corrosion/muck inside that needs cleaning out.

For the MOT you don’t need the bike started or running provided the battery gives enough power to show the lights/horn all work.

Best regards,
Richard

I’m guessing your bike has a foot change, is it the lever on the front of the gearbox, or the right hand (outside) of the gearbox that is stuck? My reference to pawls is for the right hand side foot lever mechanism. The lever on the front of the gearbox just rotates the selector shaft and that moves a single fork into one of 4 positions (1-N-2-3) so no rotating barrel and pin type arrangement. If the box has had water in the sliding dog can rust onto the splines it moves on. It is not a long job to remove and strip the box just remember the kickstart bronze outer nut that takes the kickstart spring is left hand thread. Also heating the end cover makes removing it a lot easier as a bearing has a press fit in it and must seperate to split the end cover off the box.

Cheers,
Richard