No name boutique or Big 4 TAS to get to private equity?

This is my first WSO post. I'm finishing my sophomore year at an SEC school with a 4.0 GPA and a double major in finance and accountancy. When I was in high school, I didn't even know what investment banking or private equity was, and planned to start in Big 4 auditing, switch to private accounting, and work my way up to CFO. After getting more experience with accounting courses and even starting an accounting internship, I realized that my personality doesn't fit accounting and that I wanted something more forward looking and fast paced. I discovered IB and PE about a year ago, and I decided to change career goals (IB -> PE) a couple of months ago.
I've been using my network and cold-calling to get phone calls with several lower MM's and a couple BB's and trying to gather as much info/advice as I can.

Obviously, I'm late in the recruiting game and I'm coming from a non-target school. I think I have a good chance of getting into a lower MM bank, and BB/EB isn't completely out of the question. Several students from my school have landed EB and BB positions, but most of them had two SA positions and I only have time for one. If I don't get an EB or BB position, and my goal is to eventually break into PE, would it be better to go lower MM or Big 4 TAS to start out?

I'm already aware of the fact that TAS and IB are not the same thing, and that over analyzing hypothetical situations won't get me any closer to my goals if I'm not putting in the work. That's not what I'm asking about in this thread. Ultimately, I'm just wondering if it would be worth my time to recruit for Big 4 TAS as a backup. It's already likely that I'll have to re-brand with a top tier business school, and I don't think Big 4 TAS -> business school -> IB would be a far leap and it would add a well-known name to my resume (even if it isn't the best route). Several of the BB bankers I've spoken to started out in auditing (targeted them because they can relate to my story), so I know it's possible. The accounting department faculty love me, and my school is a target school for Big 4 (Big 4 firms are on campus almost every week).

To wrap it all up, would the name recognition of Big 4 TAS outweigh the actual IB experience of a no-name MM if my long-term goal is PE? If you've taken either of these routes or know of someone who has, I'd greatly appreciate your advice! Thanks