I always like to start w/ the basics: compression test, basic electrical, etc. If you have poor compression, then it will be next to impossible to get it to run correctly.
You shouldn't have a rich condition w/ the new carb if it is set up right. The stock carters are notorious for plugging up & conking out at idle, but the MCs will idle down to 500, smooth as silk.
The ICM is a possibility, but they usually just go out.
Fuel pressure should be fine if you kept the return line to the tank (that bleeds off excess pressure).
Have you checked all the grounds in the engine bay? You may be getting a very weak spark at speeds not high enough to trip the alternator into charging. I'd freshen up all your contacts on the battery, cables & ground wires...especially the ones connected to the engine & charging system.