I'd suggest taking the old belt to a good automotive parts store such as NAPA. They might be able to find a suitable match for it.
Your suggestion sounds good to me.
A man only needs a belt that works, not one made for that mower. I used to work at a NAPA warehouse, and they carry a multitude of belts of different lengths, thicknesses, and shapes. There ought to be something close enough to work.
I remember an "emergency belt kit" that contained a belt and a belt joiner (the joiner connected the ends of the belt, kind of like the removable link on a bicycle chain). You could cut the belt to the size you needed, and join the ends with the joiner. That let you carry one belt that could be made the size for the alternator, or the size for the water pump--whatever you needed.
And, incidentally--your mower sounds very much like the one I just bought, that would not start, after long storage in a garage. If I can't get it to run soon, I'm going to have to get my money back, or the fellow who sold it will wonder why I'm holding on to it for so long.