No, you can't claim one single penny of mortgage interest on SCH E because the rental property is not used in any way, form or fashion to secure the loan. You're also not investing any of the loan money back into the rental property since there is no "cash out" money to do that with. The new loan is secured only by your primary residence. Therefore all of the interest can only be claimed on the SCH A. Furthermore, due to SALT limits (State and Local Taxes) as well as mortgage interest limits, you're limited to deducting the interest on "only" the first $1M of the outstanding balance on the new loan, and a maximum of $10K for your SALT deductions.
What you may "save" in interest, you will "pay" in taxes a few times over.
Additionally, it doesn't make sense to refi any loan if the interest rate isn't "at least" 2% lower. If the new loan results in a "savings" of 1% on the interest rate, you'll never recover or even break even with the refi and closing costs you'll pay. Think this through long term. Short term savings now will cost you dearly later.
