The wattage on R3 does not have to be equal to the peak watts. The peak current is only for a short time.
Here's a Switch Mode strawman [untested]:
Speaking of the inverter, this isn't going to work as already pointed out - the voltage on A0 is negative. It also removes your control over Q3 as it will now need a negative voltage to switch off.
Your scheme of "run for a bit then stop to check" is not as precise.
We still have no idea what the supercap is for. What is it powering? But we can have some fun speculation while we wait for the OP to tell us why these suggestions won't work.
Hi All,JCA is correct: Q2 must be connected correctly so that it can turn ON!Also, it would be better to interchange the values of R6 & R7. Now,look at the gate of Q1: R5 & R9 form a voltage divider so the gaterests at 6V while U2:A is off. The Vgs(on) will keep the super capfrom taking a full charge. (How can messed-up schematics likethis be published?)Herb