To remove the pulp: be prepared to get messy! You can do this a number of ways, but I simply peeled off a couple sections of the skin (which is hard to do if you have really ripe fruit), dug the seeds out, then used a spoon to get the remaining pulp. See the "Recipe Notes" section for a link to the Ohio Pawpaw Growers Association suggestions. You should have about 1-1/4 cups of pulp.
Add the pulp and remaining ingredients to a blender and process until fully pureed, about 1-2 minutes, scraping down the sides as necessary. Taste and adjust seasoning as needed (you may want to consider an extra pinch of salt or lime zest). Pour into popsicle molds (I use the Zoku Classic Pop Molds, which hold about 3 oz each) and freeze overnight.
Place all ingredients into a blender and puree until fully combined, about 1 minute, scraping down the sides of the blender as needed. Refrigerate in an air-tight container for 5-7 days or freeze for 3-6 months.
I made this sauce a bit on the tart side to stand up to the fattiness in the rest of the dish I served it with. If you want a more balanced sauce, I would start with 2 tablespoons lime juice and increase it from there.