MBB Interview Offer, worth taking with new job?

Would appreciate some honest feedback on this. Normally I'm pretty good at navigating this kind of stuff, but I'm honestly unsure what to do in this situation and would appreciate some advice from anyone with more experience navigating this.

I'm a recent college grad about to start at a "upper Tier 2" (I hate that terminology, but you get it) office. Think LEK, Kearney, Oliver Wyman. Very excited about it! Set to start in mid-August.

I just received an email offering me a first round at an MBB firm in the same city (big market city) to start in September - I must've applied to ages ago. I'm very, very torn. 

On the one hand, I could never get through this process quickly enough to receive an offer before I start, so I'd be going through a series of interviews where my options are 1) not get an offer or 2) get an offer, and either i) decline it because of timing, and feel bad or ii) accept it and cause a massive stir. Feels like all bad outcomes. Besides, I'd have to spend the last few weeks of my final summer doing case prep. 

On the other, it feels silly to forgo a better opportunity, with better compensation, better exit opps, etc. over a company that I haven't even worked at (I interned elsewhere last summer). Part of me will feel silly if I try to exit to MBB in two years, when I could have done it right now. Why not give the interview a shot and figure it out from there?

Would love to hear from anyone with a little more corporate experience. My main two questions are: is there a notable enough prestige difference between "upper Tier 2 firm, big city" and "Tier 1 firm, same big city" to warrant any of this? And, secondly, how much of a stir would leaving a job after one month/ two months cause? I come from the IB world, where this would ABSOLUTELY be wildly unacceptable, but I've heard some conflicting things. 

Thanks! Any and all input appreciated