First thought is that it might be a partially clogged fuel filter. It's not affected by heat, but can be almost clogged, have crud fall away while it's not running so it seems ok when it first fires up, then pick up more crud & get more clogged again while it's running. So it might seem like it's getting worse because the engine warms up, but that might be just because it warms as it's running longer. Symptoms of a clogged filter typically show up under higher fuel demands, like driving uphill or at higher revs. And I think a lean engine runs hotter, so it could be that restricted fuel is causing the higher temps, not the other way around.
Because you talk about it acting up when hot, another thought is vapor lock. When the fuel is heated it can boil & turn to vapor in the fuel line. Mechanical fuel pumps can't pull vapor, so it doesn't run. Typically it's a complete shut down, not a "hardly runs" situation, so this seems less likely. Cures include re-routing fuel lines away from hot exhaust or installing an electric fuel pump back at the fuel tank. Back at the tank fuel is cool & liquid. If fuel in the line turns to vapor between the pump & carb, the rear pump continues to push cool fuel thru, which clears the vapor & still delivers fuel to the carb.
Another vapor lock issue is fuel boiling in the carb itself. Cures are reducing under hood temps, fixing cooling issues so the engine runs cooler, & adding a phenolic spacer under the carb to separate it from manifold heat.
