Hey everyone - Just registered here to offer some thoughts on the rough cold start issue. I've been running and tuning a 2016 392 Hemi in a 2008 Jeep Wrangler, using a 2010 Dodge Challenger PCM. The engine has a Ram 5.7 timing cover and intake manifold, with a throttle body adapter that adds enough intake volume to make it comparable to a 6.4 Hemi Ram intake manifold. It's a clean and tidy package, but definitely a franken-hemi combo haha. Anyway, I've seen the exact same problem with cold starts: very rough, wildly fluctuating RPM from about 1 second after firing up, with near-stalling. Aside from the older PCM and intake, the engine is stock.
I've been logging cold starts using the AEM 30-0300 X-Series wide band O2 sensor and all of the other relevant vehicle data. It's clear that most of the startup/warmup settings in the stock tunes are inadequate to prevent a seriously lean condition that coincides with the rough running while in open-loop. The engine needs a lot more fuel in the first seconds after startup, before going into closed loop when the narrow band O2 sensors / fuel trims take over. The Hemis up to at least 2016 use two major levers to influence startup/warmup fueling: "Fuel-Air Cold Enrich" and "VE-ECT Multipliers."
The FA Cold Enrich is a table that adds varying amounts of fuel to the base commanded fuel-air ratio according to the startup engine coolant temperature. This table is only active for the first eight seconds or so, and tapers to zero in that time.
The VE-ECT Multiplier and VE-ECT Factor tables are a correction to the base volumetric efficiency table. The VE-ECT Multiplier table is active all the time (even at normal operating temperature) and varies according to engine RPM and manifold absolute pressure. The VE-ECT Factor table applies a multiplier to the VE-ECT Multiplier table and the Factor varies by engine coolant temperature. Generally, the Factor table has higher values at low temperature, making the VE-ECT Multiplier table have a larger effect at low temperature. (confusing much?) Basically, these two tables (along with the FA Cold Enrich Table) can also be adjusted to "correct" the volumetric effficiency, so the computer can adjust fueling based on a more correct amount of calculated air in the cylinders.
If you've gone round with a dealership and eliminated the other possibilities (bad fuel injectors, sensors, maybe even a re-flash) and still have the problem, I'd recommend finding a local aftermarket Hemi tuning shop that can adjust those startup tables. The more I've studied stock 5.7, 6.1, 6.2, and 6.4 Hemi stock tunes and played with the tune on my setup, the more it seems like the Mopar software engineers cut corners and half-*** these tunes. Alternatively, you could purchase your own tuning package (I'm using HP Tuners but there are others that can make these adjustments). Either way, you (or the shop) would also be able to correct other annoying quirks with the tune.