I've lately been trying to tune a Midnight Sons/League of Assassins deck, and a question has come up. If my teamup doesn't show up, can I use Stick's ability to meet the activation requirements for, say, Tower of Babel or The Demon's Head?

Stick's effect lets the targeted MK character be the target of plot twists as though it had all affiliations; that's as far as the benefit goes, though. His effect doesn't mean that you control a League of Assassins character for something like Tower of Babel, and it doesn't mean that you have an LoA character you can exhaust to pay for The Demon's Head's cost.