What if we organize the transaction in a different way: if you're the receiver, the sender of the transaction only signs his input, sends this information to you, and you use an additional input controlled by yourself and sign the TX? So you would have the control over the txid. This would result in a bigger transaction (in bytes) but still could have advantages over an approach with two transactions. But I don't know if Bitcoin allows that ...

Possible problem: If the sender knew when you exchanged the commitment transactions and he cooperates maliciously with your channel partner, however, he could try to instantly spend the output in another transaction hoping it gets confirmed first than your transaction. This would be however a relatively risky attack.