Stuff a rag partway down the drain pipe, and fill to the top with expanding PU foam (also known as spray foam). It's used in construction to fill gaps, such as between door frames and walls.

Is it ideal? No. Is it a pain in the ass to remove? Yeah, kinda.

But it expands and hardens to a watertight closed-cell foam plug, which is what you want. If you have a hardware store nearby, it's relatively inexpensive.