Closer to the end of the book, Sheymov's contact from CIA refused to go with his plan to fake their family death when defecting. In Chapter 11, "Misha" refuses to help with the plan:
Victor nodded, then introduced his embellishment. “Good. Now, Misha, I will need your help on something else. We’ll have a tremendously greater advantage if the KGB is not aware that I’m in the United States.” Victor paused. “How about leading them to believe we’re dead?”
Caught completely off guard, Misha stared at Victor, trying to see if he was joking. He wasn’t. Misha chuckled: “Look, let’s just keep this professional, OK? We don’t have time to fantasize.”
“Misha, just listen to me. I think we’ve got a real opportunity here. For example, the three of us could go to the countryside, rent a rowboat, and capsize it. If it’s abandoned near a large water system with a fairly strong current, it would be perfectly natural not to find our bodies. I know for certain that in a number of such drownings every year the bodies simply never turn up.”
“No, Victor. Now you listen to me. Without mentioning the fact that a stunt like that would rob us of at least eighteen hours’ lead time, I can guarantee that nobody in Washington is going to approve it. Our highest priority is to deliver you safely, without taking any avoidable risks. So just forget it, and let’s concentrate on the task at hand.”
Later in Chapter 11, he tells his wife about this and that he's still interested in the idea:
“What did your contact think of our ‘dying’?”
“According to the Americans, ‘playing dead’is not a good idea. From a conventional point of view, I suppose I have to agree. And that idea with the rowboat has to be abandoned. We can’t pull it off without the Americans’ help. Besides, they’ve got a point, it would rob us of eighteen hours. Nevertheless, I’m still determined to try to lead the KGB to believe that we’re dead. I’m going to do my damndest to pull it off. The fact is, there’s zero risk in it. If we succeed, we’ll have a tremendous advantage—not to mention that it would make things much easier for those we leave behind.”
Yet, in the last chapter 12, it indicates that the death faking indeed happen, but I don't think it was described how. All we see is that Victor is worried about being recognized, which would destroy his plan to be assumed to be dead:
“I know, but it’s a tough call. Uzhgorod is in the border zone. On the one hand the Border Guards will every now and then raid trains like this one. They ask everyone for full identification and want to know exactly where you’re going and who you’ll be staying with. That’ll blow up everything I’ve been lining up so carefully. Also, our being here would be registered, and good-bye to any hope that they’d count us dead.
The last "About the author" section confirms this:
In 1980, after establishing a contact with the CIA, Victor and his wife, Olga, and their five-year-old daughter were exfiltrated from Moscow. The KGB were even persuaded that they were dead. For his contribution to the national security of the United States, Sheymov was awarded the CIA’s highest medal.
So, what exactly did they do to pretend to be dead if the plan with the rowboat was rejected as it needed help from Americans?