Exclude Time Machine backup from "Open With" menu
You should be able to exclude Spotlight from adding the apps on the Time machine drive to the "Open With..." menu by opening System Preferences -> Spotlight -> Privacy. Then drag the icon of the Time Machine drive into the "Prevent spotlight..." window. This adds that drive to list of drives that Spotlight should ignore for searching operations.

Note that you can't prevent Spotlight from indexing that dive as Time Machine depends on Spotlight to keep a catalog of files, but if you do this and then rebuild the Spotlight index it should prevent Spotlight from adding apps on that drive to your "Open With..." menu.