Am I being an irrational and stubborn girlfriend?

Whether you're being stubborn or not, I agree with you! I'm just as stubborn lol! I'm so stubborn that my boyfriend pretty much spoils me. On the other hand, his family is very important. I can see his reasoning for getting upset. BUT, this is your health you're referring too! So much can happen, not just including sun burn that will effect you from staying in the sun so long. He has to take into consideration your thoughts as well. If that's a problem, maybe the trip isn't necessary now? If you do happen to go, I'd say yes, don't make it out to seem like you're totally irritated! It's the first time you're meeting his family, make a good impression doll! Because your first impression is the one that matters!