DS, this sounds at first like a bum rap, but do not take defeat without a fighting! In every place I have ever been, there are always those that back-stab others, sometimes they are successful, but not always!
This sounds like a case where you could really show how professional you are. Plus utilize your connection.
First of all, did you accually fill out an application for this position? If not itis not unusual at all to submit a cover letter/resume with an application.
Even if you have already filled one out, you can submit a resume in addition, either directly or through the existing instructor.
In this submission, what you do NOT do is mention these awful backstabbers. What you do want to include is a bit about your methods of instruction and safety measures you utilize when instructing others.
Make it clear without going overboard, that you take the safety of those you instruct seriously and follow all safety precautions such as:
Wearing helmets at all times when working around horses
checking all tack for wear areas prior to every student use
checking saddles for correct cinching before & after mounting
providing all students with a written list of rules
etc, etc, etc
You of course include references that they can contact.
That takes care of that side, and once you get an interview you can include things like you ride bareback on occasion when training your own horses, but never allow students to do so.
As for the ones that are deflaming your character, if they have really submitted, in writing, that you are unsafe only because you ride bareback when "schooling" your horses, a visit to an attorney may be in order. A nice official letter from an attorney should stop that immediately! Many professional riders ride bareback, some even bridleless too! That does not make them unsafe!