KYSU Pawpaw Planting Guide
The godfathers of pawpaws say, "Pawpaws are ordinarily quite difficult to transplant. They have fleshy, brittle roots with very few fine hairs. Experimentation has shown that, to be successful, transplantation should be done in the spring at the time that new growth commences, or soon after. (This is basically the same as for magnolia.) If many roots are lost, it may be desirable to prune the top to bring it into balance with the remaining roots."
Your potted ones should be fine if you wait until the leaves have grown a quarter inch of so. I've never had any trouble in the summer - just did about 5 in the last week or so. Flawless, even tho bare rooted (and one of them lost a tap root). Most did not even wilt, and some had growth showing like 2 days later.