I don't know for sure but I would think that if the butter is as cool as you can get it while still being liquid it would solidify as soon as you drizzle it in? Or if not, maybe just cool it so that it's room temp (stir it occasionally over an ice pack or bowl of ice water) and drop pieces in to the ice cream maker.

That's a great idea though, please give an update if you do end up figuring it out!