It's straight forward, but I would suggest it's most likely a waste of time and money to do so.

It's not very likely that the breaker is the issue, but it is possible. If you throw the main, then only the lugs where the feeder is connected will be hot (and not protected by a over-current device). To me, the most risk while working in the box is loose ends of wires flopping around and coming in contact with the lugs. Replacing a circuit breaker is a pretty simple operation, and reasonably safe as long as you pay attention to what you're touching.

replacing the double breaker was easier than I dared hope and the AC seems to be staying on this time. I do notice that this new breaker has warmed up to 86 F over 90 min, using my laser thermometer to measure the heat. Normal or is something not so good going on? All the other breakers are smaller but only a GFI breaker is as warm as 85 F.

Should have spent the $$ toward an amp probe instead. Then you would know how much current the ac is drawing. That is what you should be more concerned with. Fixing the problem, not the symptoms.

breakers are thermal-magnetic.

They use a thermal section to provide a slow trip for short overloads, and a magnetic trip to quickly clear large faults.

Yes they get warm.

The IR thermometers are notoriously inaccurate unless adjusted for the emissivity of the surface being measured.