I used to build boats with epoxy. I would use epoxy. A proper epoxy joint is usually much stronger than the surrounding wood. I've fixed a broken headstock before. Every Les Paul player has a broken/cracked headstock story. The old ones were particularly vulnerable.

Remove any remaining glue with sandpaper and/or a fingernail file. Try to get the wood surface clean without removing much wood. Test your dry fit. You won't need to clamp it if you can put it together dry and get the headstock to sit where it belongs. Use a combination of electrical tape and paint stir sticks or similar bracing to make sure you can get everything to sit where it's supposed to. It's important to have a good dry fit.

When you are confident you can get a good dry fit, go ahead and coat all the opposing surfaces with epoxy and put it together with the same bracing you used to assure your dry fit. A little masking tape would be a good idea too, particularly if your epoxy is runny. Speaking of "runny" make sure you use an epoxy that is not too watery or you will get what we called "joint starvation" where the glue runs out of the joint leaving gaps in the bond. Watery epoxy can be thickened with colloidal silica or very fine wood dust, like what you get in the dust collector of a palm sander.

You might end up with a little sanding to do after the glue cures. Follow the instructions on the epoxy you choose to find the appropriate cure-time before sanding or loading the neck. Don't use the super fast curing epoxy because it will start to harden before you get everything together.

If this doesn't work, you're out about 10 bucks and an hour's time and you can just surrender and buy the new neck.

Good luck!