Stuck bolts!

I know this has been covered but its late and i'm trying to find a a solution quickly.

a friend does triathlons, he is going on a trianing camp tomorrow so decided to replace the broken cleat on his shoe, now unfortuntely when it was installed not a lot of grease was used, resulting in a sticky bolt, and the metal its made of is made of cheese so he has torn through the allen key fitting with an allen key it does have a slot for a screw driver as well but i dont seem to be able to get much leverage as it is quite shallow even with a big fnckoff screw driver.

does anyone have any bright ideas to resolve the situation?

informative picture attached.

Make a bigger groove. HTFU
Drill it. HTFU

Or just HTFU

If I drill it, how will I get the remaining bit of bolt out?

Drilled hardware of an old boat before but I wanted the removable bits rather than the boat so didn't map much attention to the rest of the screw?

I wonde whether melting the remiander of the cleat away and then using a big pair of grips might work...

You dont need to get the rest of the bolt out. If youre using SPDs, or something similar, you'll get a new back plate in the packet, so drill the heads off the existing bolts, watch it fall apart, and replace everything from scratch
Something like this?

Or maybe a screw extractor? Reverse threaded drill bits that screw into the remaining screw/bolt counterclockwise, and drive that bolt out as they tighten

^i'm assuming they are plastic cleats, if he plans on melting them.

maybe use a hacksaw to cut a large slot in the head of the bolt and use a wide screwdriver?

yep sorry for lack of picture but they are look keo cleats, i think the idea of sawing a bigger slot is worth a go then attack with an even bigger fnukoff screwdriver