1988 Dodge Aries Dies when hot and won't start until it cooled off again

Your description suggests two different things. It's stalling after ten miles and the starter won't crank the engine. Both of those things could be related to the charging system and the battery is running down. The parts you replaced are in all different systems and will cause different symptoms. Have you checked for spark right after it stalls or won't it crank either? Have you checked for diagnostic fault codes?

Chrysler had extremely little trouble with Engine Computers. Code 55 just means "end of message". All of them end with that. Keep in mind that if there were other codes, they got erased when the battery was disconnected.

Intermittent stalling is commonly caused by the Hall Effect pickup assembly under the rotor in the distributor. It was common enough that many people carried spares in the glove box. The next time the engine stalls, check for spark. If it's missing check if the automatic shutdown (ASD) relay is turning on during cranking. The easiest way to do that is to measure the voltage to the ignition coil positive terminal. You should see 12 volts for one second after turning on the ignition switch. What's important is if that voltage comes back during cranking.

I saw you replaced the pickup already and should have mentioned that. About 90 percent of the people use incorrect terminology or don't understand what they're talking about or were told so it's just easier to cover all the bases even when it means repeating some things.

My daily driver is an '88 Grand Caravan. I ran into a similar problem two years ago and it was extremely intermittent until after a year that was the first time in its life it didn't get me back home. In my case it would stall on hot days at anything over 1/4 throttle. It didn't act like loss of fuel pressure so assuming it was loss of spark, I used a stretched-out cotter pin under the hood to bypass the starter relay to check. By the time I opened the hood, unplugged the relay, and pulled the coil wire, I had spark again and the engine would start. After a year of this, when it failed completely it turned out to be the ignition coil.

You can do the same thing. Your starter relay will have four terminals with possibly five wires. Two will be fatter. They're red and brown. One of the smaller wires is yellow and there may be two of them on one terminal. Unplug that relay, then jump the red and brown wires to crank the engine while you watch for spark.

Don't worry about the ASD relay. They rarely fail, but when they do it's usually due to corroded contacts and they aren't intermittent.