There are a couple of options IMHO:
1- if you use multiple accounts, you could create one for the gift card. To keep it out of your Unallocated balance (if you want to) you’ll need an envelope to match. Then you would pay the bill as usual but from the gift card account, then transfer money from the gift card envelope to the one you pulled the bill from.
2-if you don’t want to track the card, I would also pay the bill like always, then add a credit to the checking account for the amount (like you’d do if you returned an item to the store, for example). This will keep your checking account balance accurate and your bills all tracked, but your envelope won’t reflect that spending.
3-or like you said, you could just delete the bill for one month, but I’d invariably go looking for it, forget I’d used the gift card, and panic that I hadn’t paid it!
It all just depends on how granular you like to be in your reviews.
Hope that helps!