You aren’t paying for the merit badge per se, you’re paying for the physical manifestation of it. You have to do the work and meet the goals in order to get the merit badge. Once you’ve earned the merit badge, there’s no need to pay for the embroidered patch if you don’t want to. It’s not going to affect whether or not you are able to get your Eagle.
…No?
You aren’t paying for the merit badge per se, you’re paying for the physical manifestation of it. You have to do the work and meet the goals in order to get the merit badge. Once you’ve earned the merit badge, there’s no need to pay for the embroidered patch if you don’t want to. It’s not going to affect whether or not you are able to get your Eagle.