so once it gets to 1/2 tank.
after it cools down will it start and run. IE doe it need to have a full tank to run.
If it won't run after it gets to 1/2 a tank. even the next day I'd say you have a fuel pressure problem.
and at a full tank, there is enough fuel in the tank to help the fuel pump keep up with the engines demand. but once it gets below 1/2 its loosing the battle.
most riding mower have a fuel pump thats operated via the crank case pressure. looks like this
https://www.amazon.com/CRAFTSMAN-RIDING-BRIGGS-STRATTON-REPLACEMENT/dp/B01I85TG9Y/ref=asc_df_B01I85TG9Y/?tag=hyprod-20&linkCode=df0&hvadid=241885289460&hvpos=1o4&hvnetw=g&hvrand=398210105582602102&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=1027027&hvtargid=pla-409610154873&psc=1
between that and the fuel filter would be my trouble shooting go to....
now if the mower will start and run just fine on 1/2 of a tank after its cooled down, but maybe you are getting some vapor lock from the heat of the engine..