A substitute for carrageenan?
I doubt very much that you'll be able to substitute any amount of arrowroot for carrageenan. Arrowroot can substitute for other starches, but carrageenan is a gum.

If you want to be able to melt the cheese easily, gelatin is your best bet, and you should be able make a 1:1 substitution (although the process is obviously different - you need to bloom it first, be careful not to overheat, and give it plenty of time to set).

On the other hand, if it's meant to simulate a hard cheese (which is likely if the recipe calls for kappa as opposed to iota carrageenan) then agar-agar is probably a better choice; you'd use about 1/3 of the amount of carrageenan (or less) and, again, it's a different process, you'd basically need to boil it for a few minutes to get decent hydration.

If you're of the mind that gelatin is not vegan (it's technically an "animal product", although it's so heavily refined that the definition is practically meaningless) then you can try the agar-agar substitution, just don't expect it to be exactly the same.