I met this need by taking a cage-type water bottle to the store with me when I shopped for an insulated cup. I found one for $4 at a TJ Max store. The cup has no handle, is the same diameter as a standard bottle, and has a dimpled rubberized housing that keeps it from rattling or coming free unless I pull it out. I had a choice of about five different cups in the same price range that would have worked.

More recently, I've seen several at a Target store that looked to have exactly the same shape as a standard bottle, except for the opening at the top. They even have a thick band of rubber where the cage would hit.