I may be doing my taxes incorrectly, but I'm pretty sure you can deduct your mortgage interest on your rental income Schedule E as an expense during the year. Or, if you create a business, whether sole proprietor or otherwise, you could deduct it on Schedule C as an expense.
To me, this created a loop hole of sorts as I mortgaged my main home to pay for the rental home in cash (for a better interest rate on the loan) and can deduct the mortgage interest on either Schedule A or Schedule E.
I would also second onlykelsey in that there is someone more qualified than I to tell you about the tax implications, but from my experience, if you fear rising interest rates by June 2017, refinance now and let that cash reserve grow somehow in the interim. Then, as you said, run the numbers again in June and see what's best. I will add that getting a new mortgage might be more challenging and may come with a higher interest rate if you are carrying the 80% loan from your current home.