Does Bank Prestige Matter? Need to make a decision

Currently have SA offers from a decently known MM bank and a smaller MM/LMM bank. The smaller firm seems to have a great culture + good comp, whereas the decently known MM bank I've heard can be a grind sometimes. Smaller firm also seems to have true opportunity for Analysts to have a big role and for promotion, but I've also heard the experience you gain isn't as valued because of the lower-sized deals. Broadly same industry groups.

My question is: does the prestige/brand name really matter? I am pretty set on getting my MBA after my analyst or associate years and exiting to a field like corp dev. Are M7 MBA programs going to care about the firm I worked at or will they value my story/experiences/GMAT/GPA more? Will I also be able to exit into corp dev at a well known company (ex: fortune 500, FAANG) if I choose the smaller bank?

I appreciate any advice or thoughts. I understand getting MBA's don't really matter as much anymore, but I have no idea if I want to go into corp dev or go into VC/growth or just exit finance completely so I want to have an mba as my current goal for the foreseeable future. Thank you