There's no reason the Hue Bridge shouldn't be able to be "paired" with multiple third-party apps/services at the same time. (It's not really pairing like it is for Zigbee or Z-Wave; it's just registering that app with its own unique Hue ID/"username" for authorization to the API.) I'm sure there's a theoretical or practical limit somewhere, but lots of people have probably dozens of these without problem.
As for the problem on the Hubitat side, I'm not sure. Hubitat should listen for an SSDP broadcast from the Hue Brdige in order to discover it, which I think the Bridge sends pretty frequently, but the Hubitat integration app does advise you wait a few minutes. I'd definitely do that just in case. If you haven't tried this already, you could try completely removing the Hue Bridge Integration app on Hubitat, then reinstall it and see if it discovers the Bridge then. Rebooting Hubitat (don't go crazy with this, just once, and do it politely from Settings) or the Hue Bridge also can't hurt, though I see you've tried at least the latter, and maybe even your router just to be extra-sure? 
I also wrote a third-party integration, CoCoHue, that lets you just specify the IP address (so you'd need a static/reserved address for the Bridge), but I'm not out to plug that as a solution here when the built-in integration should work for you, too.
