Hi,
Since you are already trying the fuel cap trick, I'll give you some other ideas/things to check.
1) Altitude: If you live anywhere that's not close to sea level, your jetting could use some adjusting. When the engine pulls down, see if choking it slightly causes it to get back on govenor quicker, if it does, your lean which would cause the engine to run hotter than intended.
2) The fuel filter being 1/4 full is not an issue, don't worry about it.
3) When the engine is pulled down is the exhaust black while it tries to recover? What color are the sparkplugs? Is the engine hard to start back up after it stalls or is it harder to start when it's stone cold?
4) Is the engine bolted down tight? A lose engine vibrates more and can cause the fuel in the float bowl to foam, when that happens the carb cannot meter the ratio properly and the mixure will go very lean. Check the bolts
5) How many hours on the engine since you last pulled the covers off and cleaned out the chaffe? Most air cooled engine makers say at least once a mowing season, more in dusty conditions. If this is a Kawasaki or BS Vangard V-twin, pay close attention to the area between the cylinders, they collect a lot of debris there.
6) Take the hood off. Nothing like letting the engine have free and open access to the air. Some older lawn tractors did not have enough air flow with the hood on, simply taking it off solved the issues.
7)How old is the fuel? If your using winter blend fuel in hot temps, with a hot engine, you could be boiling off in the floatboal and causing the same condition as frothing/foaming fuel noted above. Get some fresh fuel and see. Also, fuel stabilizer does effect how the fuel performs at high temps, if your fuel is heavly dosed with it, that may be part of the issue.
Honestly, anyone with gasoline powered equipment should be using VP racing fuels small engine fuel(SEF), it contains no on the road EPA additives and does make a difference in the performance of these small engines.
Small Engine Fuel (SEF?) From VP Racing Fuels
Since this only happens when the engine has been running for some time, then happens with greater frequency the longer it is run, I would look to cooling and venting being a large part of this issue.
Of course it's been 30 years since I ran a small engine repair shop but these are the steps I would take if you came in and told me everything you have already tried.
Tom
Afterthought: Is the powerflow new to the picture with this mower? If it is an additional load, and the engine isn't at 100% either from hours/wear or improper setup, the additional load may be more than it's able to output and it's not able to catchup to the load. Mowers should be run at about 75-80% of capacity, the small gasoline engines don't have the torque to catch up like a Diesel does.