The opening is too small for the pump head. Had to use metal shears and cut some lines in it and bend back the "petals".
The car is running great. Under full throttle, it would kinda ratchet up in power, in steps, not smoothly. After the fix it is smooth and more responsive.
The little filter on the bottom (circular, very fine white mesh) looks to be the culprit. Since I was able to clean it, I put it back, knowing that I can always get at it again if I need to. From looking at the design, gas flows into the plastic chamber through the screen and past a check valve. Its looks like only the force of the liquid level does this, as the plastic chamber is not sealed. So when the round filter clogs, no gas makes it to the pump. With over half a tank, the fuel can into the chamber from the top, that's why the problem is only evident with less than 1/2 a tank.
I considered leaving the filter out, but since I can get at it now, I figured it will keep some dirt out.
Interestingly, there was very little dirt or debris in the bottom of the tank. I didn't need to clean it. Guess it was all caught in the little filter.