Your plan definitely sounds doable, mainly because the maximum mortgage limit for Nassau County is $726, 525 for 1-Family Dwelling, $930,300 for 2-Family Dwelling, $1,124,475 for 3-Family Dwelling, and $1,397,400 for 4-Family Dwelling.
Your 3.5% down payment will be based on total loan amount, which includes purchase price + rehab costs (remodel, consultant fees, title update fees, inspection fees, contingency reserve, etc.). Example: $400k purchase price + $75k rehab costs = $475k loan amount x 3.5% down payment = $16,625 down payment.
Work with a Certified 203k Contractor and a lender whose 203k experience can be verified. It's crucial to your success to select partners based on their verified knowledge/understanding of the 203k and not because they would have the lowest fees/costs.
There are 39 Lenders in your area that have done at least 1 FHA 203k Loan during the previous 12 months but there are a few of them that have done more than 20. So, the good news is there there appear to be many options for you regarding the lender selection.
Not that many Certified 203k Contractors available, though, but that's okay, because it's the HUD approved FHA 203k Consultant that determines the cost of the project and not the contractor.
Also, may want to consider negotiating a 45 day closing time, but check with your selected lender first for their recommended closing time.