Put the original TIPM module back in. There is no telling if the used one is any good or not or even how long it sat on a shelf. The unit can be bad and still test good.
Make sure the mechanic is using a new Mopar camshaft sensor. I had the exact same problem you did and it was a bad camshaft sensor. It would run ok, then rough, then turn over but no start.... A failing sensor starts acting up as it warms up and causes issues until it completely fails. A bad sensor will also not send signal to allow spark from the coils. A completely failed sensor will not allow it to start even for a second as there is no spark for combustion.
A compression check is good to guage the overall condition of the engine itself, but I can tell you even with low compression in 3 cylinders it would start and just run like crap.
It sounds like this mechanic is just guessing or is overwhelmed with work and just throwing parts at it..... Put a brand new Mopar camshaft and crankshaft sensor into it. Check the connectors for broken wires/contamination. Also put the original TIPM back in and I bet it starts