The transaction fee for a refund transaction wouldn't be an issue (based from the context) if you have to return to him the bitcoins that he have sent.
Or you could also use it as an opportunity to CPFP his transaction: just set a higher transaction fee to the refund transaction so it can bump his transaction.
Just make sure to spend the correct UTXO (the output of his transaction) by utilizing "coin control" of your wallet.
My item is on the care of escrow which we are both waiting for this transaction to confirmed before we can all move on. The problem is he doesn’t want to bump the fee while the bump feature on my wallet for the incoming transaction doesn’t have this option.
Appreciate all the reply everyone. I will try to ask first the person I’m dealing with if he wants to post public our transaction. For the mean time, I will lock this thread since I knew that there’s no way to completely remove this stuck transaction.