Humm that's a tricky one. An insurance rider may not help you in that situation. I have a competition handgun I travel with that is worth about $4k, I got a rider because I travel alot to matches. It is covered worldwide, however there are some exclusions, Like if I leave it somewhere unlocked. Like an unlocked car. SO you really need to speak to an underwriter and make sure your exact situation is covered. Different companies are differnt. For instance most insurance companies including the NRA's. WOnt cover a gun if it is lost or stolen while flying. Farm Bureau did though. So you really need to check. You could always buy a fire rated gunsafe and keep the saddle in there. Wouldnt get stolen and may survive a smaller fire.
Id be surprised if the BO's insurance would cover property that doesnt belong to them.
These are some things that you really need to discuss with an insurance underwriter.