No Name PE firm internship - do big banks care?

As the title suggests, I currently hold an internship at a no-name PE firm in college out of Nebraska and have been networking pretty hard for Pub Fin and Corporate Banking roles I have applied to at various banks. I would love to go straight to IB or DCM but the path that other alums from my school have taken is usually CB to DCM to other roles at large banks since my school just doesn't have great brand. Also, I am just very interested in CB or Pub Fin to start since it seemingly doesn't immediately pigeonhole you, and debt is interesting to me. 

Anyway, will my PE internship award me any merit, or do areas of large banks like CB and Pub Fin not really care about that kind of experience? Is it all based on networking?

I apologize if any of this seems uninformed, I joined the finance party late and I am currently a junior trying to secure his future lol. I appreciate your input, whatever it may be!