I am on the last step of building an escrow system. The user has a shared 2x2 address, and the user knows both public keys and both private keys.

How can they build a transaction to move money from the shared address to somewhere else?

My current solution involves sending the user to https://coinb.in/multisig/ to try and build the transaction, and that works, but it is not a very stream-lined process.

Electrum is able to find unspent transactions associated with an address without having to download the entire blockchain, so I know that this is possible.

Whoever answers this question best will get the prize after I get it to work.