Incentive-wise, however, in the case the sender is a service like an exchange, which must be profitable and thus "just work", there is no reason for him to "cheat" changing the transaction ID,
For the general case I tend to agree, but it is too big an assumption here because of how bad it can get.
and you would also not lose anything beyond transaction fees.
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.
The only way out of this is to beg your channel partner to be nice and sign a new refund tx for the real tx.
Both the spender and your channel peer have a leverage on you.