No! You would loose the complete access to the utxo, as in this case you comitted to a multisig with your channel partner [which you don't trust and] which provided you with a "refund" transaction... Not valid anymore (as it spends a non-existent tx). As this "refund" transaction is actually the first commitment transaction, neither the channel can operate.
Ah! I think you're right, I didn't take into account that the receiver of the tx loses control completely over the funds as the sender is the only one signing the transaction which transfer the funds to the channel's "multisig address".
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 ...