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Engine Rough Idle During Initial Warm-Up After Cold Start and Diagnostic Enhancements for DTC P0507 (Reprogram PCM) #05-06-04-053 - (Aug 25, 2005)
Engine Rough Idle During Initial Warm-Up After Cold Start and Diagnostic Enhancements for DTC P0507 (Reprogram PCM)
2006 HUMMER H3

with 3.5L Vortec Inline 5 Cylinder Engine (VIN 6 -- RPO L52)

Built Prior to VIN Breakpoint 68125462

Condition
Some customers, with vehicles built prior to VIN 68125462, may comment on a rough idle condition starting 5 to 10 seconds after a cold start when the ambient temperature is in the range of 10 to 26°C (50 to 80°F). This condition may last for up to 30 seconds before the idle speed will smooth out.

Correction
Reprogram the PCM with the latest TIS calibrations.


Only bulletin I could find with reference to rough idle.
Now, if the new program also overwrote the TCM functions, the transmission has to relearn the transmission adaptive pressure, and this can take a few miles. Not sure how this would affect idle, but it could.
As for the battery, the PCM does monitor the charging system, and if there is a dead cell the generator should be on full charge more than usual, but this may not actually set a code.
Hopefully, before they replaced the battery they did a diagnostic check of the engine charging system.