My understanding (which comes from working at a payment solutions provider), is that travel companies that apply a surcharge for payments from credit cards, identify those cards based on the BIN (the first 6 digits of your card number).
Since the Monzo cards have a debit BIN (you can check it here) -
I wouldn’t expect you to be charged for using your Monzo card.
Having said that, we’ve seen issues with companies not recognising Monzo BINs as debit in the past so I’d recommend double checking this with the support team via the in-app chat.
If you do, please do let us know what they say 