How would a mortgage lender evaluate an existing private mortgage with no monthly payments?

I could qualify for a $200k mortgage (with 20% down if needed)

Presumably, that's 20% of $200k, which is only 6.7% of the total value of the house. So in all you're borrowing $560k on a $600k house. That may affect how they consider your application.

The bank won't see the $400k as part of the down-payment. It's borrowed money that you owe to somebody else.

There's not a way to just "transfer" a mortgage from one property to another. The current mortgage will need to be released when you sell the house and a new mortgage established. Otherwise it's just an unsecured personal loan.

Most likely you'll need to refinance with the private lender as a second mortgage. The bank will want to be the primary lender to ensure that they get their money back in case of foreclosure.

So it's up to the bank as to whether they'll want to take the first mortgage, and up to the private lender as to whether they'd want to take the second. I would note that this is significantly more risk for the private lender since they're second in line (and not getting any principal back in the meantime). So I wouldn't be surprised if they weren't interested in refinancing the interest-only loan.

A simpler solution may be to just refinance as a single mortgage and have your parents continue to gift you some amount (perhaps the new interest amount?) if they want to. Any principal you may goes back to you as equity in the house.

My debt to income ratio will be very low since I make no monthly payments on the private mortgage.

Yes but it's somewhat artificial, and the bank may see through the unusual situation.

As a side note, if you can't afford to pay principal on the current mortgage, how are you going to afford to pay any principal and interest on the new mortgage? It sounds like you're getting yourself in a very dangerous situation.

I agree with your assumption and I think you should have no problem getting a bank to approve a mortgage in this situation. From the bank's point of view, you are only taking out a loan for $200K on a $600K house, and the bank still gets the lien on the entire house, so there is nearly no risk for them.

When determining your ability to pay the mortgage, your income is balanced against all of your current debts and monthly payments towards those debts. The fact that you have an agreement with your parents that you owe them $400K isn't relevant to the application if you won't ever have a monthly payment towards that loan, unless your parents actually take a lien against your property, in which case the mortgage likely would not be approved (though, as long as the bank gets the first lien it still might).

Note that your situation isn't much different than if you didn't have the current property and your parents just gave you $400K as a gift to put down on the new $600K property. The key point about gifts for down payments is that banks want to make sure it's truly a gift and not a loan that would require adjusting the DTI (debt to income) calculation, so they may require signatures from the gift giver to prove it's truly a gift. But in your case, even though it's technically a loan instead of a gift, DTI would never be affected. Furthermore, the bank cannot lose a dime unless you default on the new mortgage and the value of the $600K home drops below $200K, which is extremely unlikely from an underwriting scenario. I believe this is a shoo-in. The complexity of your specific scenario is mostly just noise that shouldn't matter for securing a loan.

As a side note, you already have a 67% down payment. You won't need an additional 20% (of any number) to secure the loan or to avoid PMI. The 20% rule is only relevant in regards to the purchase price, not the loan amount. In other words, since your down payment is (much) more than $120K, there won't be any PMI.