You likely have a tuning issue given that the plane can't go from idle to full throttle quickly, but the symptom you're describing sounds to me like a tank issue. If the plane flies fine until it gets to half a tank, it's most likely that the fuel line has come loose inside your tank. Either that or it has a split in it that lets it leak air once it's not submerged in fuel anymore.

For your advancing the throttle issue, get the high end set properly which means a 300-400 rpm rich. If you're using the pinch test that will be a barely perceptible increase in rpm when you pinch the line. Then go to idle, and listen for about 30 seconds. If the RPM drops over time, you're too rich. It it increases or the engine suddenly dies at some point, you're too lean. Get that right then recheck that you're still slightly rich on the top. These will give you a preliminary setting that will be close, but may not be exactly right. The next step is to see how the engine behaves when you slam open the throttle. Sputtering indicates that the idle is too rich, while hesitating indicates that it is too lean. Adjust accordingly. Then do the same test the other way- run the engine wide open for 10 seconds and slam the throttle closed. If it dies you're either too lean on the idle or you have your idle set too low. The last test is to check the mix with the nose pointed up. Have the engine at full throttle and hold the plane up. You should hear a slight RPM increase and the engine should hold that increase for 10 seconds or so. If it goes up and then back down you're too lean on the top. Tweak the needle until it does that, then go richer by one click at a time until it holds the rpm increase. Go ahead while you're at it and drop the throttle closed after that 10 second run to see if the engine will die after the lean nose up run. If it does, you need either a richer low end or a higher idle setting.

If you go through all of that, you should get 100% reliability of the engine in all maneuvers and never have to worry about a lean run. Keep a mental note of how the engine reacts to the pinch test after you're all done so you can replicate that same top end mix when the weather at the field is different. If you're inclined to invest in the hobby a bit, a tachometer is a really valuable tool for learning to tune engines. It takes the guess work out of the rpms your getting, letting you repeat your adjustments more precisely. Over time your ear will get tuned to RPM changes so you won't need the tach as much, but even then it's really helpful with a problem engine to be able to put numbers to what you are doing.