I figure my options are to sell and take a loss (not ideal), or preferably find coverage that allows or doesn't care about the K&T.

Some carriers offer loss assessment insurance, which can come into play when the HOA experiences a large covered insurance loss, such as a hail storm requiring roof replacement or a fire requiring building replacement.  

We're at a loss on what to do here.

These self employed 2nd mtg's are typically based on alternative method of caluclation like profit and loss statements if you're self employed or the last 12 months of bank statement deposits using a formula to derive an income from these deposits (like 50% of all annual rolling 12 calendar month deposits X 50% /12 months = monthly income).

Would you prefer they change their policies just to fill your rental quicker and then increase your risk of losses?

You could lease part for food service or coffee shop and some retail space, a small health club, massage. 

Short sales are where the bank would realize a lesser loss settling than they would completing a foreclosure.

Or the market tanks and the value of the house goes down and that creates issues (selling at a loss, who brings money to the closing table, etc.

Defaults and Remedies A default exists under this note if (1) Borrower defaults in the payment of this note or in the performance of any obligation in any instrument securing or collateral to this note; (2) (a) Borrower or (b) any other person liable on any part of this note (an "Other Obligated Party") fails to timely pay or perform any obligation or covenant in any written agreement between Lender and Borrower or any Other Obligated Party other than as described in (1) above; (3) any representation in this note or in any other written agreement between Lender and Borrower or any Other Obligated Party is materially false when made; (4) a receiver is appointed for Borrower or an Other Obligated Party or any property on which a lien or security interest is created as security (the "Collateral Security") for any part of this note; (5) any Collateral Security is assigned for the benefit of creditors; (6) a bankruptcy or insolvency proceeding is commenced by Borrower, a partnership of which Borrower is a general partner, or an Other Obligated Party; (7) (a) a bankruptcy or insolvency proceeding is commenced against Borrower, a partnership of which Borrower is a general partner, or an Other Obligated Party and (b) the proceeding continues without dismissal for sixty days, the party against whom the proceeding is commenced admits the material allegations of the petition against it, or an order for relief is entered; (8) Borrower, a partnership of which Borrower is a general partner, or an Other Obligated Party is terminated, begins to wind up its affairs, or is authorized to terminate or wind up its affairs by its governing body or persons, or any event occurs or condition exists that permits the termination or winding up of the affairs of Borrower, a partnership of which Borrower is a general partner, or an Other Obligated Party; or (9) any Collateral Security is impaired by loss, theft, damage, levy and execution, issuance of an official writ or order of seizure, or destruction, unless it is promptly replaced with collateral security of like kind and quality or restored to its former condition.

A 1033 exchange is specifically for involuntary conversions and catastrophic loss situations like yours.