Issue with shop/warranty for 2007 Hyundai Tucson

Hi car folks!

So my dilemma is this:

My Tucson had its engine replaced in 2022 by a small local shop. The engine replacement came with a 5 year or 50k Mile Parts & Labor warranty. I have the invoice from said replacement and am well within the set parameters.

Almost a month ago, my Tucson broke down on the highway. Had it towed to the same small local shop, they said it was the timing belt and that we still had the warranty on our engine. Okay, cool. Except the shop manager cannot find the warranty info. She didn't put the company on any of the invoices and can't seem to find any of the paperwork/emails/anything. For almost a month this woman has been "looking" for my warranty paperwork, or anything that could point her to where the engine came from.

For three weeks+ she's been ducking my calls and when I do get ahold of her she gives me "I haven't had a chance to check this other storage area yet" and the cycle continues.

Today I marched my ass across town on foot and showed up, asking for a plan. Her plan is to have her guys replace the timing belt and "see if that works" and to my understanding, chances are low of that being enough. She refused to entertain the idea of not finding the warranty and assured me she will find it.

Tl;dr- shop lost my warranty on an engine they replaced, it busted and now they can't find the warranty info to utilize said warranty to fix my shit.

What are my options here? What if she can't find the warranty info? Is this normal??? Any help is appreciated, I'm losing my mind.