Due to his girlfriend, I'm not going to bring my buddy to a festival.
If you don't want to bring someone you don't have to. Its up to you in that regards.
The girlfriend will get pissed off if he's not home
Your buddy has a love life so it is his relationship that bothers you. If it bothers you that much then why not just tell him to come alone?
I agree completely, I just don't want to cause problems with his girlfriend.
I'm glad you agree and I'm sure your buddy will appreciate the invitation. Explain to him and his girlfriend that you just want to hang out together as friends and he can spend time with her afterwards. She'll survive.