the 2.5mm heat shrink should work for you, but it depends on how you do your splice. however, if you just wanna cover where it solders to the board, 2.5mm, or 1.5mm should be fine on 20gauge wire.
My father in law always told me about Harbor Freight... if you use the tool every day, don't get it there. But if you use it ocassionally, then it should be fine.
when i think of heat gun, the brand Ideal seems to come to mind, but they might be a bit pricey.
heat gun vs butane torch, might get you a heated debate on the realms of Mac vs PC, or IE vs firefox. I dont trust myself with something i could burn myself easily on, So i'd go with the heat gun, yeah, i can still burn myself, but at least the house stays safe : )
EDIT: i have thought about the $13 heat gun from Harbor Freight, I ended up getting a Drill Press instead for now.