It's not a public road, so most of the usual legal wrangling ("same rights and responsibilities") that we usually pull out won't cut it. This is more of an office politics issue than anything else.
1) Find out what policies, if any, actually bar you from riding a bike on the perimeter road. People enforcing the "rules" should be able to point out what rule it is that they're enforcing.
2) Whether there's a real policy or not, contact whoever's in charge of employee health, environmental issues, or similar. Most large employers have at least one. If there's a policy, you're trying to get them to change it. If there's not, you're fishing for a memo clarifying the status of bikes on airport property. Escalate if you get no response.

In general, I would expect that anywhere you could bring a private vehicle, you should be able to bring a bike. I would be surprised if there's anything actually barring you from biking on the perimeter road. As others have pointed out, there's no way there can be a security issue with a bike that doesn't exist with a car, and if you're not in areas where there are aircraft, bike safety is no different from any other road.

However, even from a public perspective, airports are usually amongst the most bike-unfriendly facilities imaginable. So do what you can, but don't be surprised if you are eventually just knocking your head against a wall.