Very nonstandard question: Is there any way to widen a tire by combining it with some other tire (cut and re-glue)? Brace yourself - this is not normal.

I have an old 22" rim (37-501, 22x1-3/8) and all I can get is skinny tires for it. But I'm an idiot and I really want a wider tire on that rim. I want it so bad I am grasping at straws for ideas.

So here's the thought:

Get a new tire (22x1-3/8)
Get a new BMX 22" tire (which is NOT gonna fit my rim)

Cut the skinny one's beads off with about 1" of rubber above it.
Cut the bead off the BMX tire very low.

Glue BMX tread to old-school beads.

Viola! a 2" wide tire for the old 22" rim.

Yeah, I know this is nuts. But CAN it be done and become whole again?

Maybe, just maybe, someone here knows if this has ever been done. Please let me know.
I don't think what what you are proposing to construct would not be anywhere near strong enough to function properly as a tire.

Tires are not just molded rubber. They are contain a casing of continuous fibers that enables then to withstand inflation pressures and lateral forces caused by cornering.

https://en.wikipedia.org/wiki/Bicycle_tire#Construction

By gluing parts of two tires together you would not have a continuous casing running through the tire.