You may be willing to pay a higher rate on the construction loan if you're doing construction-to-permanent financing and can get better mortgage terms or a longer, better rate lock from that lender.
"The Best Strategy for Converting a Construction Loan"
You should take out a mortgage to pay off the construction loan. Construction loans aren't meant to be a method of long-term financing. A first mortgage is a better choice than a home equity loan because you can borrow for longer periods, generally at a lower interest rate.
A typical home equity loan is a second mortgage. It carries a higher interest rate than a first mortgage because there is more risk to the lender. That's because the first mortgage has to be satisfied before any sale proceeds go toward satisfying the second mortgage. With no other lender having priority in the event of foreclosure, you've given the home equity lender all the benefits of being the primary lender.
Ask yourself, "What's in it for me?" It comes down to the loan terms. You should get better terms on a first mortgage than you would on a home equity loan.
"Converting a Construction Loan to a Permanent Loan"
The advantage of this type of loan is that there's only one loan application and one loan closing. The idea is that the lender finances the construction of your home, and when it's ready for occupancy, the loan is converted from a construction loan to a mortgage.
You pay a price for that convenience. You are a captive customer and don't have a lot of negotiating leeway when presented with the interest rate on the permanent financing. Decisions about the loan term and fixed vs. adjustable mortgages are made when you close on the loan prior to construction. You're also assuming a financial risk that the house is well-built.
A one-closing loan will typically be priced with interest rate locks that limit the interest rate on the permanent loan. Alternately, the loan can have a float-down option that will let the borrower take advantage of declining interest rates. A float-down option can have an interest rate floor that limits how low the interest rate can go.
If you can get a better rate by finding a new lender, then refinancing can make sense. Whether you convert to permanent financing before refinancing depends on how long a window you have before the construction loan has to be converted.
If you can avoid paying the $350 conversion fee, you can use the money toward closing costs on the refinancing.
Review your loan and any rate lock agreements to make sure there aren't any prepayment charges, minimum loan term commitments or other charges or penalties. If there are, then the decision to refinance becomes more complicated, and you should hire a real estate professional to help you evaluate the decision.
If it's clear sailing on a refinancing, then use a refinance calculator along with an estimate of the closing costs on the new loan to determine how many months it will take for the monthly savings to pay for the closing costs. (This is also known as the payback period.)
If you only plan on being in the house for a few years, you shouldn't be willing to refinance with a loan that has a long payback period.