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I know nothing of Paw-Paws but have made a number of fruit-infused liqueurs. With a fresh, ripe fruit, you don’t need a high proof spirit or the very lengthy infusion times as often recommended for limoncello. 
Choose your base spirit based on your intended use and personal preferences. Vodka will be most neutral and let the fruit flavor dominate. A white rum or blanco or reposado tequila might also be nice. 
I’d recommend this recipe for tequila por mi amante as a guide for you to use with vodka, rum or tequila, as you like. Use the paw paw flesh without skin or seeds in case they have any bitter flavors. Or leave them in to see what happens. 
In any case, the infused spirit will taste smoother after it sits a while after straining. You can sweeten it at this time, as one would do with limoncello but I’ve found fruits like berries or melon to be pleasantly sweet without added sugar. 
Edited to say that you can add some citric acid and malic acid for a bit of tang if that’s needed.