That could cause severe damage to the house without a structural engineer and probably a very expensive contractor. Are you finishing the ceiling as part of this project? If so, it might be short-sited to do this for a extreme short throw projector. If not, I would consider whatever you have temporary since you might want to redo the space in the future to be “finished”.

I would recommend either putting the short throw on a small cabinet underneath the screen, getting a slightly smaller screen, or looking at a standard throw projector swap. All of those would be cheaper than hiring a structural engineer to facilitate moving ceiling joists.