Lien holder won't allow property to be rental; Refi options?

[Eric Boring]

Hi everyone,

My wife and I are just getting into real estate investing. I have read Brandon Turner's book on Real Estate Investing and his book on Rental Property Management. We are moving in two weeks and want to use our current home as a rental rather than selling it. The problem is, our current lien holder does not allow the property to be used as a rental. We've talked to a bank, two mortgage lenders, and a credit union about refinancing and each of them have told us they require us to have in savings a reserve of 6 months of the mortgage payment, plus 2 months of payments for our primary residence (amounting to a little over $12k for our situation). They said this is standard among most lenders because Fannie Mae and Freddy Mac have set those rules. They also said they will scrutinize two months of bank statements to make sure there are no shady deposits or borrowing short term just to meet the requirement. We have about $4k right now and I calculate it will take us about 3 months to build up the rest. I don't like the idea of the house sitting vacant for 3 months while we wait for the refi. My questions are:

1. Would a private lender be a good option for the refinance?
2. Are there any options or recommendations I'm not seeing?
3. How would the current lien holder discover we're use the property as a rental? How likely is it they would find out?