Would transferring to a better school in the middle of a graduate program irreparably hurt my professional connections and academic reputation?
This is fine. Don't feel guilty at all. I say this as a math professor at a mid-ranked state school, who has written rec letters for students who, for various reasons, wanted to transfer out.

Be sure not to speak ill of the place you're leaving, but you owe it to yourself to seek out the best opportunities available. Good luck!