Legal Question in Real Estate Law in California

First, unless the house that is rented and upside-down is in a completely separate entity, e.g., owned by a corporation or LLC, you can't just put it into bankruptcy. Bankruptcy doesn't encompass particular properties, and you can't just select Property A to put in bankruptcy; it is entity-wide or person-wide.

Next, note that short-sale deals can differ in respect to the lender having recourse after the sale closes. Lenders like to slip in clauses giving them future recourse to the selling borrower's other assets. You or your agent should try to negotiate these out.

You didn't mention whether the rental income is covering your expenses. I assume it doesn't, or you wouldn't be considering drastic measures, but negative cash flow can be as much an issue as being "under water" on a debt/equity scale.

You also didn't mention a foreclosure option, but it would be useful in planning what to do about the rental to consider letting it foreclose as an alternative to the other three possibilities (keeping it and enduring negative cash flow, short sale, and bankruptcy). There is a set of statutory rules governing whether a lender can come after you for a deficiency after a foreclosure sale fails to produce enough cash to fully pay off the lenders. Generally, under these rules, purchase-money firsts and seller-carried seconds are relatively safe from actions seeking recourse following a deficiency as a matter of law.Even where a borrower might be liable for a deficiency, many lenders these days are simply foreclosing by trustee sale rather than going to court to foreclose, which they must if they have a deficiency judgment in mind.

Finally, a second-mortgage lender that loses its collateral when the holder of the first forecloses will become an unsecured creditor of the borrower, and can sue directly without being limited to recourse to the collateral (which is gone).