Yeah, the "SS is a Synth" would be a fun twist if ANYTHING IN THE GAME SUPPORTED IT.
Which doesn't because, well, the Vault Tech rep remembers him.
Also, the SS clearly DOES remember things before the bomb dropping since he knows about baseball.
OK, I really, truly hate to say this but: implanted memories. We know it's possible due to Nick. And an identical body. The problem is that I don't see any reason for Shaun and the Institute to do this. But let's handwave that for now.
I'd prefer a Blade Runner-like hint setup where there's some vague evidence spread around to suggest that it might be possible, but it's up to the player to decide.
While I feel like the whole "SS is a synth" idea is a little trite, it would be dramatic to realize that hunting down and killing Kellogg, and everything else, was all done for a fantasy. It also somewhat reverses the role with Shaun, because you're his father, but you're also his son. And it brings in a lot of philosophical questions about whether a person is more than their memories. If you had all of the SS progenitor's memories, are you the SS, a cheap copy, an improved version or something else?
Granted, that last part is identical to Nick, but it's different when experienced from the player's perspective. And those issues weren't explored very thoroughly with Nick.
Suggestion after these ruminations: Provide hints and clues throughout the game to get the player asking questions. Spread them out Morrowind-style so you won't easily get the whole picture on a single playthrough. Have a point near the end where it's revealed that the SS is a synth. Make the player sweat it out and consider it. Then provide contradictory evidence after the end. Leave it up to the player to decide what the truth is.
Plus, you can drive players crazy if there are scattered hints that point one way or the other. People would go mad exploring everywhere to find the answer, when (hopefully) there's just more lore that ultimately leads nowhere (in terms of an authoritative conclusion). I like it.