What is my liability regarding the sale of a house with a problem being handled outside of the closing?

Don't do this.

1) If  you make an agreement with a buyer to pay a certain amount or have certain work done, as a condition of sale, that deal is generally enforceable, even if it is not reported to other parties, like lenders. You don't indicate which state you are in, and in some states, contracts for amounts over certain dollar limits (e.g. involving payment of more than $5k) have to be in writing to be enforceable; so it *may* be the case that in your state, an oral agreement for a large-enough amount is not enforceable. It may also be the case that in your state, if the agreement is considered one for the sale of real estate, it would not be enforceable if it is only oral and not in writing. (Though its not clear that this "side deal" would be considered a contract for the sale of real estate--you could easily end up in costly litigation over whether it is or isn't.) Look up the "statute of frauds" for your state, or repost your question with state-specific detail, to see what agreements have to be in writing to be enforced. The corollary of the above is that if the agreement is a written one, it will almost certainly be enforceable.

1a) If the agreement is enforceable and you do not honor it, you could be sued for its value; or it may possible give the buyer grounds to rescind the sale entirely.

2) More  importantly, it may be that if the septic system was falsely reported as being up to code or adequate or in good shape, or if the money to repair the system effectively changes the sale price of the home, that you have committed fraud against HUD and the lender. This could result in liability and/or in a denial of financing or rescission of financing if it comes out.