Recently we had a requirement to search based on a particular attribute within the lookup view. The initial thought was to add a Find Columns to the lookup view. But then realized we do not have option of adding Find Columns to the lookup view.
The next thought was if I add that particular field to the list of Find Columns in Quick Find view will it work in Lookup View also? And it worked, basically adding any fields in the Find Column will make it Searchable.
The search works properly in the Lookup View as well as Lookup Inline View.
Hope it helps..