In my opinion, get a short term hard money bridge/construction loan. It's expensive, but you can close in 2-3 weeks, finish the job and refinance.
Lenders, Coney Island Capital Group, or Approved Funding Mortgage Bankers may consider it.
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A good Rule of Thumb is if you have great credit and good relationship with your bank, they are probably going to be able to give you the best deal. Banks are going to be a little stricter and there is more red tape in order to qualify. A Broker will have a wider vareity of loan products. Interview at least 2-3 Mortgage Lenders and make sure to compare apples to apples. If you can get references from past clients, by all means call them and check on the service. Remember that a mortgage is not just about the payment and the interest rate. Primarily, focus on the best loan that fits your financial needs/ goals, not only for today, always keep the future in mind.