blackhawk121:
You should make the move towards this start-up gig if you feel it is the best experience and where you will learn the most. Do not worry about what looks best in your application, particularly since you already have the blue chip name under your belt. It would be an entirely different situation if you were deciding between a blue chip company and a non-blue chip company for your first job. Then, brand name might have an impact.
As long as you have a credible and logical reason for switching and believe you can (and will) achieve tremendous things at the new company, you should go for it. That will ultimately come through in an application more successfully than taking a role primarily on the basis of name.
Lastly, I am less sure about the benefits of having time to work for your family business on the side. It could actually work against you as it will make the startup job seem less demanding. I do not know the exact parameters of how this would work out, but my guess is that does not leave a lot of time for extracurricular and outside leadership. You will need that to round out your application, unless the family business is your main non-work (startup) thing.
Also, why do you need new domain expertise? That might be useful as you think about a long-term post-MBA career but it might create unnecessarily complications in your application. Diversity in industries could potentially send a mixed message to the AdCom, unless there is some overlap in function or industry. Just think about how that would all play into a cohesive application.
Good luck. If you are considering using an MBA application consulting service, please reach out to us as we are happy to help.
https://admissionado.com/Best,
Seth