Lancelot, I only worked the 737-3/4/500 series for a short time, and it was over 20 years ago, so the memory is a bit foggy.
That being said, I'd be suspicious of the PMC. The CFM56-3 PMC was very crude - an analog device (rather than the digital PMCs on the CF6-80A and CF6-80C2). In fact, it wouldn't surprise me if the CFM PMC is the last time Boeing put analog electronics on a new aircraft. Anyway, it could be the PMC is acting up since it only comes into play at higher N1s.