The main pitfall nobody's mentioning: even if you win in small claims court, you get a judgment, but that judgment is just a piece of paper. Your cousin could ignore it, and then you'd have to go through the separate process of trying to enforce it - which might mean garnishing wages, placing liens on property, or hiring a collections agency. That costs more money and time, and if he's already ghosting you, he might not have assets worth pursuing anyway.
That said, the handwritten agreement should absolutely hold up. Judges deal with these constantly and as long as it's clear about the amount, when it was due, and both of you signed it, you've got a solid case. Small claims courts exist for exactly this situation. The filing fee varies by state but usually isn't huge - maybe $100-300 or so depending where you live.
Your best move is probably to send him one more formal communication (email works fine, texts less so) laying out exactly what you're owed, referencing the agreement, and giving him a deadline - maybe 10-14 days. Keep it businesslike, not angry. If he still ghosts, file. It'll put pressure on him and at least you'll have an official record. Just go in with realistic expectations about collecting afterward - sometimes the threat of a judgment is enough to make someone suddenly cooperative, but sometimes it's not.