@Luke Jacobsma I can only speak from a private lender's perspective as my experiences are limited to that space. There are others here way more qualified with regards to conventional financing and its regulations/underwriting guidelines. I believe portfolio lenders may be the best ones in terms of being open to working with you since they hold onto their origination.
With respect to non-bank lenders, the rate is usually not negotiable. Lenders base their rates on their underwriting guidelines and their rate sheet. This is something that is set and usually updated monthly to account for market conditions. If you pre-qualify or are pre-approved for a certain rate, that's usually because taking into consideration your characteristics and qualities as a borrower per their guidelines you fall into that rate bracket or tier. If for example you have a 650 FICO, are looking for a 70% LTV loan, and have 2 years experience and this falls in their tier 3 which shows a 4.5% rate but they then issue you a 3.6% rate because you're going back and forth negotiating, this would be an exception to their guidelines and would need to be documented with an explanation for the exception.
It's not good practice to make exceptions, especially because these lenders usually then sell their loans to investors and investors buy their loans after a review of their guidelines.