One more angle on the "Fritz dies, Katte survives" idea: what does Wilhelmine do? Because the relationship with her brother was so absolutely quintessential to her life that I think one would be justified to let her do something radically different in such an AU. Such as running away herself. England (or at least Hannover) would have been an obvious destination; she was probably too much of a realist to expect being regarded as marriage material by the cousins under those circumstances (i.e. without a a dowery, and without being male, the crown prince and thus a useful political tool), but still, it was reasonable to expect them to take her in and give her a place at court.

Now, her pride wouldn't have dealt well with being the poor relation (especially at a court where she'd been raised to believe she'd be the future Queen), so she might not have remained long, plus maybe she deals with the terrible loss of Fritz by feeling vengeful. And FW prides himself on being the Emperor's loyal subject. So Wilhelmine decides to go to Vienna and make her case, accusing her father of point blank murder. There's medieval legal precedent here demanding a public hearing at last. Again, she's probably aware that this will not end in the Emperor doing something to actually punish her father, but the public embarassment as the Prussian royal family's dirty laundry is shown in front of the world? In great detail? In a world where dignity and representation means so very much? This is her avenging her brother.

And who should be another witness but a Prussian officer in exile, one Hans Herrmann von Katte.

Also, while she's in Vienna: of course she meets the two arch duchesses. Befriending ensues!