For the interest paid divided by the interest rate method, what do you think '... You can use this method if at all times in 2023 the mortgage was secured by your qualified home ...' means?
As for the 'Statements provided by your lender' method, the statement 'You can treat the balance as zero for any month the mortgage wasn't secured by your qualified home' doesn't apply for your primary home for the months the mortgage was not held. This statement applies only to the months the mortgage was held and not secured by a qualified home.
This is what Pub 936 says and it is reasonable when you have one mortgage for less than 12 months or two mortgages for all 12 months. Your situation is an example of when the average balance methods in Pub 936 are not reasonable and would, in my opinion, tweak the sold home/bought home solution I posted earlier.
For the year you either sold your first home and bought another one OR you bought a new home and made the first home your second home, it is reasonable, in my opinion, to calculate a 12 month average for the mortgages held less than 12 months. The tweak is that you don't have to zero out the months for one mortgage in the months that overlap.
You can use the interest paid divided by the interest rate method or the statement balance with zeros in the months the mortgage didn't exist. Even though this is not specified in Pub 936, it is a reasonable approach in this situation. But be prepared to explain how you calculated your deduction in the unlikely event someone asks.
