There used to be some projections done in Punta del Este over the summer tourist season on the main Gorlero Street. There also was an "auto-cine" on the beach, but that´s also gone now.
I have an old 16mm film projector but I suppose you mean a newer video projector, for DVDs. I doubt anyone here has one for home use, but I know a guy who owns one for rental purposes and I suppose that it would be available over week-ends since he usually does events during the work-week. I can ask him about this.
Today would have been a terrible night for this, but the weather should be changing for the better this month.