P0401 code with new cooler and valve, and cleaned intake manifold.

I posted in your other thread about the curious way your scan tool is showing the manifold pressure. It's even curiouser here.

As in that other post, I get the feeling your scan tool is not only converting the units to inHg, but also converting absolute pressure to gauge pressure, then leaving off a minus sign. Just as a quick sanity check, can you post what your scan tool says the pressure is when the engine is stopped?

If I'm right about what the scan tool is doing, a few things look weird here. The 30 inHg is too low high implausibly vacuumy for an idling engine, would be more like a reading during intense engine braking. 21 is kind of in the ballpark for a normal idle. 14.7 is rather high low unvacuumy for an idling engine, at least when commanded EGR is zero.

 

Ok, so the scan tool is only converting the units to inHg, but still displaying an absolute pressure—numbers go up for increasing pressure, down for more vacuum.

That means your normal idle pressure is a little high (not as deep a vacuum as expected). Typical for idle tends to be around 27 kPa (or 4 psi, or 8 inHg absolute, or around −21 inHg gauge). Yours settling down at 14.7 inHg absolute seems like maybe there is an intake leak somewhere?

I suspect the earlier rough running (and more elevated manifold pressure) just indicates the engine is under some load working to recharge the battery. It then settles down to a true idle and the lower manifold pressure. You could watch the battery current and see if you see a charging (negative) current during the rough-running period, that tapers off at the same time the engine seems to smooth out.