Welcome to GMAT Club and thank you for this wonderful and considerate post.
I have known a number of MBA folks who have gotten a pet in their second year when their life has gone more normal and more structured.
I think It will be awesome to have someone for culture shock and unconditional support - that's important when you go to bschool and dealing with a lot of rejection. At the same time, you would make a pretty bad host for your pet during the first month of your program. It is not uncommon to be going on trips, retreats, be busy for long periods of time and have very little sleep. Your cat won't be seeing much of you and you will be either a bad caretaker or will be outsourcing it to a daycare, which is something you wanted to avoid in the first place....
I can see perhaps someone bringing your friend over in a month or two after you settle in. I am not sure if that is beyond the point though and if it is going to break your cat's heart (2 of my mom's dogs died due to separation anxiety after she has taken 2 separate trips; both passed away within a few weeks of her coming back) so I understand the possible weight of this decision. It is super hard.... I think the best thing would be to see if someone from your family can welcome her to their home and then start training her for separation exhiety/etc. The issue with my mom's dogs was they never saw her leave overnight, and the first time she left for 10 days, it was the end of the world for them. I am definitely not a qualified adviser for pets so take this with a lot of grains of salt but perhaps your local vet can make some recommendations or resources online to help with cat attachment? Hopefully it is less severe than dogs... the unfortunate reality of the BSchool transformation experience is that it compeltely rebuilds you and to do that, it has to break you apart and then put you together in a diff configuration. That leaves little time you can use to care for others but it can also make you more human and more responsible and stronger and have less discouragement and hopefully cry much less during the nights. I would not bring your cat for her own sake (you don't want her to be a burden or you be upset that you can't take this last minute trip or you have to go home - that gets old) but if you think you will need her and value her company and she is a part of your life, this could be a good experience, just takes work