Your mortgage rate isn't really negotiated. Most of it is looked up in a published fee grid — and the grid has stairs. Landing one point of credit score above a break is worth more than any haggling you'll do.
Two buyers close on identical $400,000 loans in the same week, with the same lender, and the same 20% down. Say one locks at 6.5% and the other at 6.875%. Nothing about the houses differs. The gap comes from a fee grid neither of them was shown, and it prices off a three-digit number.
— Educational, not financial advice.