Matthew,
I remember when I had a similar problem, it turned out the valves were sticking.
The first thing (I think) you need to do is eliminate a mechanical cause of your problem. Because the problem correlates directly to the heat buildup in the engine, and because it is deteriorating (from last season to this), I suspect mechanical or ignition-type issues. Heat buildup shouldn't affect the fuel delivery system, unless . . . the float is sticking.
So, if I were you, I would try to recreate the condition, i.e., mow half the yard, and when the engine starts to die, I would (1) pull the spark plug and give it a compression test, (2) spray some ether (starting fluid) in the carb when trying to start it.
The results you expect from such tests are: (1) no compression = valve issues, (2) brief start/run = fuel delivery issues. If the problem isn't mechanical (compression), or fuel (tank/pump/filter/carb), then the issue must be spark.
I use a B&S spark tester, but a helper holding a screw driver 1/4" away from ground, or an old spark plug with the electrode removed are reputed to do the same thing -- tell you if you have a good blue spark.
If you want, you can reverse the order of the tests: check for spark, check for fuel, check for compression.
I'm betting the valves are sticking, but the coil may be cutting out.
Let us know what you find.