XJ 4.0L stalling above 1500RPM.

Hey all. I wouldn't ask, as I'm sure it's something simple, but my ex's Jeep shit the bed, and she's either selling it (hopefully to me if I can figure out how to move it the 1000km home), or scrapping it. I figure if she sells it, she'll get more for it if she knows whats wrong, or perhaps fixes it.

She drove around with the exhaust hanging by a thread for months. Finally had a shop install new muffler, but they didn't check the cat.
She threw codes P0138 and P0436, which suggests that she melted the cat. It's making the traditional "shake a can of nails" sound at idle.
I suspect the shop that did the new muffler were completely incompetent.
Jeep still drives, but starts to stutter/stall below 1500RPM.

My first guess is a melted/broken CAT, and I'm going to have her (hopefully) pull the downpipe and see if the problem goes away (in which case, maybe I just fly over and drive it home). If not, I'm thinking fuel delivery, plugged air filter, or bad plugs/wires/cap in that order. Sound about right? I covering all the obvious bases here?

OMG. She changed her mind. It stalls below 1500 unless she gives it gas. Runs and sounds fine above 1500.

Disconnected IAC just to check. No change.
Sounds like a tractor at 1000RPM which is where it's idling, but apparently runs and drives fine as long as you keep the RPM's up. Problem just sorta presented itself on the highway, while pulling into a gas station after a few hours of driving. Just started sputtering and died when she let off the gas. Started back up with a little coaxing with the skinny pedal, and she drove 300km to some families place and it apparently drove perfect. Still lots of power and ran smooth. Only problems at idle.
I'm almost thinking a burnt valve, worst case, but she's got a socket set, an axe, and a shovel, and I can't think of a good way to test for a burnt valve with those tools

OK, let me get clear on this. Does it stall with and without a load?

It sounds to me like it stalls just if she isn't touching the gas. Immediately sounds like the IAC, but she doesn't have the tools to really investigate, and me trying to walk her through it over the phone is giving me a head ache .

The really unfortunate thing is that the Jeep ran absolutely solid otherwise, and she said on the last 300km of the trip, she was still getting 22-23mpg (which is what she always gets). This to me suggests it's not a burnt valve cause there just ain't no way she's getting that kind of mileage with anything wrong.
But, without any tools or access to a shop, and she's going back to school and was going to sell the Jeep either way, she's just going to give it to a junk yard at this point, unless I can find a way to pick it up 1400km away.
Meanwhile, I have a cracked block in my rig, and could REALLY use that engine. It's always ran substantially better than mine.