Sorry to hear about that. I feel your frustration!

A brand new Ludwig P32 Butt is only $6. That's the route I would go. I know it won't be original, but what's the point of having it be "original" when, in this case, "original" is synonymous with "unusable".

Plus, the holes should line up exactly.

If you have a Dremel tool, you can safely grind a slot in the top of the screw to put a good-sized flat screwdriver in the slot and give it a good turn. I have used this method on several stuck screws over the years with success.

Use a narrow metal-cutting wheel on the Dremel.

What? You don’t own a Dremel? Buy one. It’s one of those tools you didn’t know you needed until situations like this.

Josh

I did look into buying an entire new butt plate and I saw that they're cheap, but I live in Europe, and it's like €30 or so just for the shipping and I don't want to pay 40 odd bucks just because there's a screw that's jammed, so I have to rule out that option.

Yeah, that might be the way to go about it. Those tiny little screws are made of very soft metal. And you would think a small screw would mean a small screwdriver, but no. I used the biggest flat head screwdriver I had and there was still some wiggle room. Enough to damage the screw. You gotta watch out for those screws. They'll getcha..... they'll getcha.

You can try a double ended drill bit type thing called an "easy out" which has some teeth that grab into the ruined screw head and allows you to back it out of the hole. You just need a reversible drill or a powered screwdriver that has a chuck for bits...and a reverse on it

(most all do these days). One end of the easy out creates a crater in the ruined head and then you flip the bit over and push it down into the crater and it grabs onto it and spins it out as shown in this pic:

https://www.walmart.com/ip/5PCS-Screw-Extractor-Kit-Double-head-Broken-Bolts-Remover-Repair-Tool/139934485?wmlspartner=wlpa&selectedSellerId=101056111&&adid=22222222227433325988&wl0=&wl1=g&wl2=c&wl3=513606593653&wl4=pla-1248867941644&wl5=9033307&wl6=&wl7=&wl8=&wl9=pla&wl10=328101615&wl11=online&wl12=139934485&veh=sem&gclid=EAIaIQobChMI3e3NiNvy8AIVVDytBh3PtQD1EAQYBiABEgLWv_D_BwE&gclsrc=aw.ds

That, or maybe if you can get some small vice grips to grab the edge of the head of the stuck screw and clamp down on it then you could back it out that way...but that wouldn't be the best approach because the grips will slip off and mangle everything even more.

Also, as mentioned, you could carefully Dremel an improved slot in the screw. ...or maybe just cut the whole thing out and get a new screw.

I can hear myself having an outburst over that situation. Nothing unusual about that. The neighbors will gladly share your frustration without even knowing what it's about.

Maybe take the butt off and soak it in wd40?