Bush was a Realist and long story short, basically didn't want the USSR to break up or there to be instability in Eastern Europe OTL, so I emphatically don't think he would care about any of that in an incipient WW3 situation. The emphasis would overwhelmingly be de-escalation not forward gains, even if Eastern Europe is in a state of eruption. They'd happily throw those movements under the bus in this situation to achieve a settlement.
I suppose one of the problems they faced was that a lot of the applicable footage for Bush would have come from the Gulf War, where he'd be using language which he wouldn't have used if the threat was exponentially greater. Conversley, it does mention that the coup does take a toll psychologically on Bush and his appearances in other media have alluded to the fact that could make him more belligerent.
Given it starts with Gorby being disappeared I'm not sure why they couldn't also bump off Bush, but I imagine they couldn't find enough footage of Quayle talking about enough things foreign policy wise.
Just a thought but have Bush be indisposed (maybe he's caught up in that Neo-Nazi attack?), Qualye being "unavailable for comment", and the heavy implication is that Baker's in charge throughout the whole thing?