You're right that a bad TPS can cause some strange driveability symptoms, but it doesn't really fit a backfiring issue. I assumed (maybe wrong) that the backfire is from the engine/ intake... not the exhaust. It's a major distiction, correct me if I'm wrong on that. Most backfire complaints that occur on acceleration are from the intake system.
It still sounds like an ignition problem of some sort, but I've got a few ideas for you to check out.
The ignition coils on these engines are gawd-awful strong, capable of putting out better than an inch of spark when healthy. That's the part we don't know about... whether the three coils in the pack are all in good shape. I'd like to to do an output test on all six output towers.
Pull the plug wires loose on the front bank first at the coil. Leave them loose (but connected) on the coil towers and start the engine. Grasp the plug wire about 3" or so from the boot end of the wire and pull back, coaxing spark out of the coil to the plug wire. Doing it this way is much safer than holding the spark plug wire end away from the plug.
Expect to see something like an inch or so of total spark, taking into accout the distance it has to travel inside the boot back to the metal connection.
Do the same thing on the back side, but be extra careful. You might ground a pair of pliers with a jumper wire or something to prevent getting bit.
I've seen carbon tracks develop between the coil wire towers (in the plastic) and the spark plug boots. Shooting the coil with some water will normally bring that sort of thing out of hiding, shorting the spark to ground.
The next thing is a bit of a longshot, but needs to be checked. Immediately to the rear of the ignition coil pack will be a 10-wire connector that feeds your injectors and MAP sensor. I had the good fortune to have just seen a problem on an 01 Caravan 3.3 this week, where engine heat and oil leakage teamed up to soften and weld the individual wire circuits together on the injector side of this harness.
With individual control circuits, 12v supply power and MAP sensor signal and ground circuits all melted together like that, it might only take a small torque reaction of the drivetrain to cause something to short momentarily. If it IS shorting, this is extremely dangerous to the engine controller (PCM), as shorting 12v straight to an injector control circuit or to sensor ground can kill the PCM. If you find any sort of reaction to a wiggle test of the harness with the engine running... or the engine develops a hard misfire... stop immediately and turn the engine off.
I wound up replacing all 10 wires for about 8" out of the connector as it headed under the intake plenum to the injectors and MAP sensor... quite a mess.
One last thing, concerning the A/C compressor; you're right that it's activated during A/C and defrost selections. I'm not sure if it's causing engine misfire or if it's just extremely rough due to internal issues. I've seen compressors that shake so badly that the Misfire Monitor sets a P0300 (multiple cylinder misfire). The RS Caravan has had some issues with compressor failure.
I'm not completely sure if the roughness AND backfire problems require the compressor to be engaged in re-reading your post, dmg. I'm inclined to believe it's only roughness you're feeling at that time and the backfire is probably a separate issue.
And if the only time you have backfire is with the A/C on, I'd completely discount the bad injector theory. Once again, correct me if I'm wrong.
Talk in a bit,