Marc,
Your last comments are starting to make more sense.
As you probably know, your Jeep has Chrysler's Sentry Key anti-theft system, which consists of a small chip hidden in the head of the key, and a little "radio" around the car's ignition keyhole that communicates with the chip in the key.
If someone should try to steal the car (without a valid Sentry Key), for example breaking the ignition lock, and turning it to start the engine............it'll go just like you said. The engine will typically start, but just for a moment.
If the factory alarm was armed, the alarm will sound too.
The remote starter installation also has a separate, small plastic box, with a Sentry Key chip inside, and a little wire loop antenna wrapped around the car's keyhole.
Only when the remote starter is turned on, it energizes a relay in that little box, so the Sentry Key can "show" itself to the car, enabling it to start. (The box remains off the rest of the time, so the car can't be stolen.)
I'm not a regular Code Alarm installer, but I've seen their bypass modules fail, on two occasions, in friends' cars.
They were both Ford products, but the concept is the same. In both cases, just the action of taking the car apart to inspect things, was enough to jostle that box around and get it working again, temporarily.
IN SUMMARY: My guess is that the Sentry Key bypass module in your car is only working intermittently.
HOW TO TEST: You can easily verify this for yourself, in just a few minutes, with a couple of easy tests.
1. Sit in car, turn ignition to Run, and observe all the lights. You have to be pretty fast.
Somewhere in with all the warning lights is a small red dot, a little smaller than a pencil eraser.
That red dot should come on steady for a second or two, then go out and stay out.
OK, so now you're familiar with how the light is supposed to work. Good.
2. Turn ignition off, wait maybe ten seconds or so, and activate the remote start while watching the red light.
If the red light is flashing instead of on steady, this indicates either A, a theft attempt, or B, the Sentry Key the car is "seeing" is in some way not valid.....defective, unprogrammed, et cetera.
3. Open the driver's window so you can't get locked out. Close all doors, lock the doors with the remote (so you're setting the alarm), and leave the car alone for at least 30 seconds (so the alarm fully arms).
Reach in through the open window, insert the key in the ignition switch about 80% of the way (not all the way, you don't want to activate the "keysense" circuit in the car) (and don't turn they key), and activate the remote starter.
If the remote start works consistently and correctly with the key in the ignition, you can be 100% sure it's a Sentry Key bypass issue.
SOLUTION: If the dealer did the installation for you, they should warranty all of this, as well as apologize for not getting it right the first time.
It could be something as simple as the antenna around the car's keyhole either isn't tight enough, or isn't lined up well. They are somewhat senstive to position.
Probably, though, judging from my experience with my friends' Fords, the module is likely to be bad and should be replaced.