The bridged adapter can't have the same MAC address. Two of the same MAC address can't exist in the same IP space, else the router can't route packets to them. Just change the bridged adapter's MAC address to something else.
Is there an issue with using Docker for this? Seems like an easier route.
Edit: I'd venture a guess that if in NAT mode, the MAC address might stay the same, which wouldn't be an issue. Are you positive it's actually using bridged mode?