Your school may be used to dealing with conflicts like this. My campus had a lot of music degree students, and they had mandatory rehearsals etc at night. We were used to having a few people every year that had conflicts, so those PNMs would come late and stay for part of the parties rather than the whole thing. e.g. if the PNM can only come for the last hour, she'll go to each of 3 parties for 20 minutes each. It's not ideal, but at least you get to go to all. Your panhellenic may be able to work something like this as well. We also had issues with swim meets on bid night, so swimmers rarely were able to partake in bid night festivities (we had a lot of swimmers in the system).
Also, our recruitment was not very competitive and if you missed one day of the first 2 rounds, it was okay. Yes the chapters could technically not invite you back for not going, but they rarely dropped someone for that. Not saying that would happen at your school, but it's possible.
Every school handles it differently- hopefully yours will be able to work with you. Until you speak with them directly, you won't know your options for the fall
