I have no idea what the rules are for responding on a federal level for your own employees without medical direction, but it's probably not going to be kosher to respond to 911 calls. Look at the primary role you guys are serving though. 911 responses (to medical calls) aren't what these resources are meant to do at all, and I don't know if blurring the lines of response guidelines would help with anything. Getting caught up in every rural call with an extended response by EMS would mean your primary mission could get neglected for little to no benefit.
If I were in this position, I wouldn't do anything more than first aid on a good samaritan level if I happened to roll up on something. Remember too, first response by non transport capable BLS is often overrated. If anything, perhaps try to set up a first response system in your jurisdiction for cardiac arrests only if the engine is staffed, not committed, and closer than any other EMS or FR. Just be aware that it can be very difficult to get incorporated into dispatching and the area wide EMS system.
This whole thing is going to be even more complex if you aren't recognized as EMT level as you stated. If it's just a bonus cert for hire and you happen to have the equipment without it being in your job description, I don't see it happening. I get where you're coming from and admire the mindset, but there just seem to be a ton of hurdles.