Most universities continue your previous GPA into your second degree so your overall GPA (from 2 degrees) will more than likely be used by medical schools. Of course Im not 100% but Im almost positive of that. They will then separate your overall science GPA (bio,chem, math) and non-science GPA (humanities and what not). If you are pursuing another degree, keep in mind that as long as you do really well, all those loads more of credit hourse will significantly boost your GPA. (I know from experience)
As far as the name goes, I dont know if that really matters. The MCAT is used as a sort of tool to determine how well you really probably did in undergrad. For example, say you achieved straight A’s from the big name univeristy and I acheived say A’s and B’s maybe a C or two in a small, public school and I got a 32 on the MCAT while you got a 24….the names of schools become less important. I think you get the picture.
If you think you would be better off in the long run with a second degree, go for it. I think your public school will be just fine.