[jb92563]

The 13.8v charging and the 12.4v engine off is right on the mark.

Sounds like the starter is not working well.

Sometimes the starter connection or battery terminals can corrode and cause weak starts even though the voltage is right.

Check for corrosion and clean the battery contacts if there is some.

Heat is also the enemy for things electrical so perhaps your starter is becomming marginal and it gets worse when its hot.

I had another car that the ignition and interior lights would come on but when I tried to start, it could not turn the engine over.
I cleaned the battery terminals since I noticed some of that greenish corrosion and amazingly it started right up and the problem was gone.

I had one car that continually had the corrosion problem so one day I vaselined it all after a good cleaning and that kept the moisture/battery acid off the terminals for good, but that was in the days of maintenance batteries. Should not see any acid corrosion anymore these days.