un-clutch
I don't think you can heat with electricity in that manner. For heat you need resistance and that steel is too good of a conductor...no resistance.
I'm not at all clear about that frozen part. It doesn't sound like a clutch at all, but rather a simple sliding dog similar to the lower unit of an outboard motor. It's In or Out and that transition must be made suddenly and at idle. If that's the case, and they AREN'T meshed now, and the dog is between you and the transmission, then it seems that a good whack would be driving it toward connection. I know that isn't possible because you said so already, so I obviously don't have the picture.
I'm gonna make a guess that the 3-tooth gear slides on a keyway which enables it to maintain connection with the engine. The most likely place for it to rust together would be in that keyway. Good penetrant and however much heat you can safely apply...with a little encouragement from the hammer, is the best approach based on my vague understanding. Actually, a BIG hammer would be best BUT don't swing-away. A tap from a heavy hammer still carries strong inertia energy that a small hammer doesn't. There must be bearings on the ends of that shaft so caution is advised.