The engine swap is bad advice.
Gen1's had transmission issues, always, 2nd, some 5th.
The 100k mile beater could never have a problem...and the 1k mile trailer queen's transmission would slip.
No real reason or give-aways that it will happen.
So, you've got a good chance of having the problem immediately in the next engine, or later on.
Vs, if you fix the transmission(have it undercut), then unless you just abused the heck out of it, you'de never have a problem again.
Ballpark $400-$500 for the undercut, plus fluids and gaskets and you're still under a grand...try finding a good engine for that.
If you have to add that much in labor...try finding a good engine for $2k.
It sounds like the shop just doesn't want to fool with it, which I understand, not that it's difficult, just time consuming.