this would be seriously easy todo with a linear actuator!

The only issue is the short throw projectors are seriously susceptible to any type of movement or shift.

I would use 4 heavy duty drawer slides with the actuator attached to one side. I would not use undermount drawer slides as they tend to have a bit of shift. Also you are going to have to watch your distance from the screen to the projector. You have no say so in this as these are fixed.

Goodluck! cool project!