How do I seal an unused washing machine drain?
Pour some RV antifreeze into the trap. It won't evaporate quite as fast as water, and will stop the sewer gas until the rest of this is sorted out.

A sink stopper won't do what's needed here. A test plug  goes in the pipe, which is probably no more than 1-1/2 to 2 inches. The seal is formed by winding up the screw so that the rubber seal is expanded against the walls of the pipe. I have no idea where you think you can stick a 3-1/2" stopper in the washing machine box you've pictured. You'll need to cut the hole for the drain properly if, as it appears, it's not really matching the hole in the pipe.