So when your patient goes south, and the lawsuits get filed, and the investigation reveals that you are using equipment that is not supplied by your agency, you do realize that your agency (or more likely, their insurance company) is likely to place all the blame on you (regardless of the fact that the reason the patient died had nothing to do with your personal supplies), because you were using unapproved and unauthorized equipment?
If your medical director hasn't authorized its use on the ambulance (because if he or she had, it would be on the ambulance, so by default, all non-provided medications and equipment is not approved, unless explicitly stated otherwise), why are you using unauthorized equipment? Couldn't it be interpreted as working outside your proved scope of practice?
I mean, using your logic, I could bring my personal set of irons (flat headed ax and haligan married together), because I might need to force a door to get to my patient to gain access to my elderly patient who fell (which leads to better patient care) or I can work on gaining access to an MVA before the FD arrives (again, quicker access leads to better patient care). Now if your agency provided you that equipment on to keep on the ambulance, that's a much different story.....