Hi Elliot,
Well, the search does not really make any different queries on desktop or mobile devices, so I believe the issue might be with the actual apostrophe chraracter. On the tested mobile device the apostrophe might have a different character code as the ones on desktop.
Anyways, I logged in, checked the configuration, everything was all right. Because you are using the index table engine, the search does store apostropheless versions of the keywords as well, and recognizes them. On the search settings I added all known apostrophe characters to the keywords exceptions list:
',",’,‘,’
This was if the user types in Don't breathe the plugin will actually search for Dont breathe instead. It will match, because the word don't is also indexed as dont.
I hope this helps!