It's hard to give much advice without some numbers, but what you're proposing can certainly be done. My only question would be if you have the money for a rehab on your primary residence, why not just use that to buy a second property? The amount of money required to add a bedroom plus other items is probably pretty significant and could be most if not all of a down payment.
It really is only beneficial if the rehab/ improvements that you do to your primary residence increase the appraised value by more than you spent on the improvements. Otherwise you will have a difficult time refinancing for much more than the principal you have paid down in mortgage payments.