Is there anything I can pack into my reproduction bathroom sink hot water faucet handle to stop it from slipping/spinning? The teeth are stripped and I'd really prefer to not have to replace it.
There are manufactured solutions for your problem that will work way better than glue/epoxy or a bunch of work with saws and stuff.
Use a "fit-all" handle arbor that attaches to the stripped stem with set-screws. There are dozens of handle styles to choose from.
Will it prevent you from having to find a replacement stem or replace the whole faucet? YES
Will it work way better than other options and still be aesthetically pleasing? YES
Is it simple and easy and quick and relatively inexpensive? YES
Will you need to change both of your existing handles/knobs? YES
Cut a slot in the end of the post with a hacksaw, so it looks something like a screw head. Make the slot two to three times as deep as its width (the slot kerf width, not the post diameter).
Find or buy a piece of stiff wire, usually called piano wire, that exactly fits the slot. The natural width of a hacksaw cut varies between 0.030" (0.74 mm) and 0.063" (1.6 mm) depending on the pitch and set of the teeth.
Now turn the handle over and examine the hole. Cut a piece of the wire that is just too long to span the hole diameter but just short enough so you could force it sideways into the hole where it would become permanently jammed in place.
Then place the wire in the slot and carefully force the handle into place over it. If you want to remove the handle from the post, the wire will stay in the hole, and when you replace it you will turn the handle until you feel the wire drop into the slot.
If this doesn't seem strong enough then you can make the slot deeper and stack up two or three wires.
Instead of wire you could cut a rectangle from a matching gauge sheet steel but this is going to be a lot harder to fit just tight enough into the hole because unlike the wire it will not bend. You would probably need to make a jig -- essentially a mild steel copy of your faucet post, with a matching slot -- to drive the rectangle into place with a mallet.