Deloitte or IA Bulge Bracket or mid-market IB

That's my choice. I come from a very non-target school 3.7 in Accounting and I've done my work in applying to places and doing well in interviews. I got and planned on doing both Deloitte Audit Internship for Winter 2022, and also the BB bank Internal Audit back-office internship for Summer 2021. Suddenly, I have a shot at a smallish Middle-Market IB internship position. The choice seems obvious for an aspiring high-finance guy, but I'm not sure it is. Over the initial phone interview, I asked what are the mid-market bank's placement rates for their internships. She said sometimes they hire from their pool of interns, 3 current analysts are former interns. She didn't say out of how many past years of interns. I didn't grill her on it, though. I did well on the phone and next week is the technical interview but I would like some advice on the possibility that I get that offer as well, as the BB back-office has been kind enough to extend the deadline till after the interview with the mid-market bank front office.

The first decision deals with which of the two big names and departments, Deloitte Audit and BB back-office, will pad my resume the best to get into an M7 MBA or even decent MsF school, to break into IB.

I am assuming that neither will really offer good shots directly at that world; Deloitte External Audit having to go through advisory, notoriously hard to do even if a good worker, and the few online posts of BB banks' Internal Audit divisions I get is that it's a dead-end, although the recruiter did mention that it's possible to go into the S&T division (mentioned in passing). My original post was going to end here and ask which between Deloitte and BB Bank name brands are better for masters or MBA school applications and take that path after a year or two working full-time at either.  I would still like to know your opinions on that.  MM bank possibility does change things up a bit.

With the mid-market IB front-office summer 2021 internship role that I'm assuming I'll get (presumptuous, yes), I would let go of a well-paid summer job (FT hire almost assured) at BB as they would both overlap. I would still do Deloitte in the spring; decent money and it's a backup career option.  The middle market position pays a measly $12/hr for internships and full-time offers seem sparse after the internships (see 1st paragraph). They're definitely not considered an EB, it's one location, I'd almost consider them a boutique but they call themselves mid-market M&A investment bank. Would this still be better on my resume than IA BB bank internship?  

I know this is presumptuous, believing that I will get the IB, and a long post, but I have the study guides from this site and have a good interview personality so I think I'll do fine.  The IA BB bank extended their offer deadline to a few days after the MM FO position, so I wanted this out there in so that both scenarios are addressed in time: 1) Deloitte-AUD vs. BB-IA for b-school applications, 2) MM bank thrown into the mix.