Has anyone used the rotary to etch a design on to a cup that has a handle?
Sure. With a chuck type rotary.
Chuck type is the easiest like Hank said but if you have a 4 inch Focal point lens and a not to wide image you can do it without a rotary a helpful cheat is to set your focal point half way between the center and end of the image in whatever axis the curvature is in.
Dollar tree glasses are great for experimenting with this
The other possibility is to cut a couple of collars (wood, plexi, ?) that you can slip on each end of the cup and have a large enough outside diameter that the handle can clear the rollers.
I do beer mugs all the time I have a counter wieght (just a chunk 3/4" hard wood “oak”) that I double side tape on the opposite side of the handle, it allows the mug to rotate 360 degrees without slipping, work’s like a charm.
I have this one
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