Ordered an ECM, having a problem

I have the vehicle listed above SXT, RWD. A month ago it developed a hard start problem and a random cutting out of the engine.
Hard start: When holding the key to start, the engine will stop cranking while still holding the key. It will take 5 to 6 attempts to finally start the engine.
Engine cutting out: Once the engine is running, while driving it, either during acceleration, or during cruising, the engine will cut out. Not sputter and shut off, hesitate and buck, simply cut out as if someone reached over and turned the key off.
It has been to two mechanics and a garage. It is not throwing any check engine lights. A new fuel pump was installed, no improvement. A new battery was installed, and alternator output was tested, both good. A new ignition switch was installed. The IAC valve, Cam sensor and Crank sensor were checked, all working fine. A livestream of data revealed to errors, Battery Disconnect and Incorrect VIN.

Ordered a new ECM programmed with the VIN, got the correct part number from the Dodge dealer. Installed the new ECM. After connecting the battery the tachometer and speedometer maxed out until a drive around the block cleared it out. An error scan revealed P0633 Immobilizer key not programmed ECM/PCM. The P/N the company sent does not match the dealer P/N, but is a correct alternate. There is 6 or 8 different P/N's for this vehicle, both P/N's come up for this vehicle.

Here is my question: Will this P0633 code become a problem while driving the car, or will it be something that clears itself after driving for a certain period of time? The ECM company wants me to send the part back, and issue the correct part, but that means another week of not having a vehicle, can't have that. All symptoms have cleared up, and the vehicle is running better than before. I'm more than happy to run with this part as long as it won't be an issue down the road.