What stabilizer can be used in non-dairy ice cream with alcohol?
Assuming a few things about your recipe - your coconut milk is full fat, your alcohol is a liquor at 80+ proof, it sounds like you're just adding too much stuff that doesn't really freeze. You've got a total of maybe 3/8 cup of alcohol, when just 2T is enough to soften ice cream noticeably. Ice cream made with just cream is already plenty soft, so adding that much alcohol can make it really soft. You didn't mention sugar, but that would make it softer as well. This is the opposite of the normal problem of too much water, not enough fat/sugar, making ice cream freeze too hard.

The simple solution is to just add more water, in whatever form suits you. Since you're using fruit, I'd suggest simply adding more of that! You could also try using light (reduced fat) coconut milk so that you have less fat. I suppose you could even reduce the booze - just slowly simmer it until the volume is reduced, so it'll have less alcohol, but retain flavor.

Otherwise you're going to have to dilute things - use less coconut milk or alcohol and add something more water-based that freezes harder. For example, you could replace alcohol with some kind of flavored extract, or replace some coconut milk with milk.

(The arrowroot isn't making your ice cream melt. It'll thicken it a bit, and stabilize it (keep the fat and water from separating), both of which are fine things. The stabilization is probably good, since coconut milk does separate eventually.)