When the house was signed over to him, was the full amount of the bond recorded as income?

It was not.  Only what was paid up front.

simplistic
adjective
sim·​plis·​tic sim-ˈpli-stik
: excessively simple or simplified : treating a problem or subject with false simplicity by omitting or ignoring complicating factors or details

Did these two events happen in the same year?

How much went into and came out of his BUF account?  I would start simply by reading the 45-page IRS Audit Guide for Bail Bond Agents.

 https://www.unclefed.com/SurviveIRS/MSSP/bail.pdf 

"Noncompliance in the bail bond industry was initially identified in a project conducted by one of our districts through the Examination function. There appeared to be a relatively high incidence of nonfilers, and of those returns audited, there was often a lack of adequate books and records to support income and expenses claimed.

The first indication that there was noncompliance in filing returns was observed by checking the filing records of persons advertising in the phone books. Of the names checked, almost a third were nonfilers.

Two primary issues were identified during a preliminary study consisting of the examination of the returns of 12 bail bond agents. Unreported income and the deduction of payments into the agents' reserve accounts (commonly called Build Up Funds or BUF accounts) were the two prevalent issues. Also, personal expenses were frequently being deducted as business expenses."

Your client accepted the property as payment in full (the balance), so the proceeds are the income. Selling expenses would still be allowed as business deduction. It's like a body shop working on a car that is abandoned with them. It is a discounted sale. But, as one of the long gone participants here used to say, you can't write off what you didn't write on. You don't take the full amount owed as income and you don't take the discount as a deduction. When your taxpayer works on cash basis, you only deal with what actually happened, not what was supposed to happen.