No need to swap wallets. Create a single 2of2 multisig wallet where each party holds a key. Send your funds two to different addresses of that wallet. Then you just sign a transaction transferring from both addresses to designated other ones (outside the multisig wallet). Since a "single" signed transaction in bitcoin can have multiple input and output addresses, each party can review the full transaction for correctness (50/50 split to right addresses) before signing.

The only challenge is "filling" the wallet, but doable: one party starts sending small amount to the multisig wallet, then the other party evens out. Increase amount by the amount the other party paid in total, until the preagreed amount is in.