Alright, You All Who Are Wise In The Ways Of Cycling, help me out.

I just bought a 1980-90s-ish Peugeot Triathlon. It was a little beat up and dusty from sitting in a garage for who knows how long, so I decided it would be a great way to introduce me to the delights of bike maintenance and repair.

The tires were Micheline Hi Lite Pro, 700x19c, and completely shot. I bought 700x25 replacements (cause all you kids like 'em so much) to find that the front tire does not clear the frame, and not by a little bit.
I bought a 700x20 for the front to discover it also does not clear the frame. If I pull the skewer so it's barely in the fork I can juuuust clear it (pics).

1) Is this a stupid thing to do? It makes me nervous, but I don't know if my anxiety is warranted.
2) What is the cheapest way to get a tire that clears the frame? All the 19s I found cost more than the damn bike. New fork?
3) A friend of mine is an engineer with a machine shop at his disposal, and he's getting all misty-eyed with fork-extending solutions. He is not a cyclist, nor has he ever worked with them, and I'm not sure I should put my life in his hands. Has anyone done something similar? Any resources I can direct him to?

Due to a front impact, your fork is bent towards the back.

Look for a compatible fork from the same era. Don't try to add anything to it or extend it or whatever. In fact, take a good look at the rest of the frame and see if there's any sort of bending, crack, denting on it. If it was involved in a front end impact there's a good chance the rest of the frame is damaged as well.

Oh well. Live and learn.

Yep, the wheel and tire are fine, but the fork is shot.

Before greeting a wall or car bumper the frame and fork looked more like this.

Really glad I posted, I would not have guessed it was a front impact. Thanks so much for y'alls help!

Nice looking bike, even if it is a bit "curvy."

And steering geometry.

It is hard to tell from the pic, but your downtube may be bent just behind the head tube.

Better? I'm not sure what angle is best for showing the thing I can't see with my newbie eyes.

Look for rippling in the top tube, and the down tube, about an inch or so behind the head tube.

From the photo it appears the fork sacrificed itself to save the frame. In fact the fork nicely spread the energy absorption along the length of the blades, and BITD we used to straighten forks with similar damage very successfully.

However, these days there's greater concern for potential liability, and few mechanics will straighten forks other than their own or those of good friends.

Sounds like a job for his machinist buddy.

I've made kinda crescent shaped shims on occasion. When placed between axle and the "crotch" of the dropout, they can lower the axle by a few mm.
For the OP, get the bends addressed first.

New fork. Or maybe some Pinarello stickers...