Hey Daniel, congrats on almost being done with the construction job. Although I don't yet have rentals of my own yet, I see scenarios like this come across my desk in the mortgage broker channel all the time. Here are some things initially to consider,
This is a tight deal. You'll want to narrow your value, because at $1.1, your LTV is ~85%, at $1.2, your LTV is ~79% to 80%. Being under that 80% is going to be crucial to reach your goal of pulling any $ out of the deal.
Also, because the loan is jumbo, rental loan, your rate is most likely going to be higher than that 5.5% offer at the current bank. There are banks that will do 30 yr fix commercial loans, with no DCR (cashflow) required on 1-4 units, but their rates are higher than what that bank got you. In order to qualify at many banks on the jumbo/DCR loans, your cashflow needs to be close to a 1.0, (Cashflow = PITI).
I think it'd be best to explore the 5.5% option that bank gave you as that is a great rate for a loan like this.
Hope this helps and you find a solution!