Would doing the internship make my resume/application for Business school look bad because I've "bounced around" with different employers?
No it wont, it is +ve if you get an internship as a business analyst, but seriously as an intern BA you wont get to get that much experience, even a real time BA who works along with me, although i had my fair share of BA, isnt really busy all day even if its a F500 company.
The point is you dont have to hate your previous experience, just because you dont like your employer, there are many different fields in Insurance just as in Banking Hedge Funds, International Finance and many more. The point is if you dont like one field you can go into a sub field, you dont have to make it irrelevant.
But if you feel that you want to be an BA then pursue it by all means.