The 'statement by your lender' method in pub 936 doesn't allow you to put $0 in the months the mortgage didn't exist. That statement is for when the mortgage did exist but not secured by your home.
But all is not lost. You are allowed to used any reasonable method to determine the amount deductible interest. When you sell one home and buy another in the same year, it is reasonable to use a slightly modified version of the the statement by your lender method as you suggested. For each loan total up monthly balance for Jan through Dec, with 0 for the months the loan was not held and divide by 12. Zero out the balance and interest for one of the mortgages in the months the loans overlap.
