Hi Craig, what you’re actually doing here is pairing the Hue Bridge with SmartThings, not the bulbs themselves. The bulbs are still only connected to Hue bridge, and the bridge passes the info about them and control to SmartThings.
Yup, the problem is that the bulbs can only be on one zigbee network. When you pair them to the remote, it creates its own zigbee network and effectively “stealing” the bulbs. I don’t know if anyone has successfully done it with Hue, but you would need to pair the remote to your Hue Hub first, then pair the remote to the bulbs. Once the remote has joined a zigbee network, it should use that network to interface with bulbs. Try adding it as a hue dimmer switch in the Hue app. If that doesn’t work, it may not be possible to do this with the Hue hub.