Alternatively, you could graft the branch of a pawpaw that flowers earlier to try and get the male and female flowers maturing at the same time. You might also be able to just buy some pollen. It's actually quite easy to collect. You can use a brush to sprinkle it on a female flower
How does this work in practice? Do other trees from the same species produce male flowers first, and then female? Logically, if all trees produce female flowers in spring and male flowers in the summer then the species can only survive if the seeds can remain viable unfertilized for months (which is probably not that exceptional for plants, I wouldn't know). But if the female seeds are still viable by the time the plant sprouts its male flowers, exactly how does this prevent self-pollination?
I should've just said "later" instead of "later in the year"
I'm not sure how long pollen stays viable in the environment. Some species can stay viable for many weeks but most are just a few days I think. They can stay viable for years if refrigerated properly however.