Will I fail my MBB background check?

I've been super fortunate to have been extended a third year undergraduate summer internship offer with MBB but I'm worried about failing my background check and having my offer rescinded. 

I'm an undergraduate student and I've been working with a Professor since September 2019 to Present and listed that on my resume (September 2019 - Present). However, I've had 3 different contracts under my Professor (as different research programs have started and finished at my university and thus where she was pulling funding from changed) and was unpaid for a stint (June 2020 - July 2020) when I was working at my last summer internship due to compliance issues and I wasn't allowed to continue paid research. 

The big problem is - will there be a red flag given that a) I listed my experience as continuous even though I was unpaid for a period of 2 months and b) my department title has changed a few times (since my Professor does research in multiple departments at my university)? I didn't note my changing departments on my resume since I was always doing the exact same work for my Professor but just a different department on paper given where funding was coming from at the time and didn't ever think anyone would be really looking into this that closely. My Professor can verify I've worked for her, but it's incredibly messy and I'm afraid of losing my offer.