This isn't a picture of precisely what I was describing since your ceiling is drywalled already. So imagine the ceiling is drywalled. Then apply the clips and channels. 4 clips, 2 channels.
Then attach a piece of MDF or plywood. You could router the edges up pretty. Attach the panel to the channel. The panel will move up and down like a shock absorber. This should help. At least it's an easy trial.
Not thinking sistering or otherwise stiffening will affect change in the small zone around the projector. The rest of the floor system will still be oscillating even though the small area around the projector is stiffened.
A resilient mount will have a better chance to separate the projector from the ceiling.
Projectors are mounted on ceilings decoupled with clips as a common practice. But the resulting spring has a low resonance due in part to the mass of double drywall hanging on it. To your point, Bill, a heavier panel attached to this little clip assembly would ensure a low frequency resonance.