The TAP error has nothing to do with your router: it means that Dolphin doesn't see the network bridge at all, as if you don't have the TAP adapter installed. Did you install OpenVPN itself? If so, that's not necessary; you only need the TAP adapter. First, I'd suggest completely uninstalling OpenVPN. Grab the latest x64 OpenVPN installer, but don't install it. Instead, open 7-Zip and use it to explore the OpenVPN installer file. One of the folders will have the tap-windows.exe installer, so extract that file and run it instead of the main OpenVPN one to get the TAP adapter without any of the unnecessary OpenVPN stuff. OpenVPN also has a tap-windows.exe listed on its download page, but I think that one is older than the one that's bundled in the OpenVPN installer.
Once it's installed, you should have a second Local Area Connection in your Network Connections portion of the Control Panel, so click and drag to highlight both LAN adapters, right click, then bridge them again. You'll lose the ability to configure the settings on your regular LAN adapter, so you'll need to right click on the bridge to set up your network correctly again from there. How you set up the BBA within Dolphin also depends upon whether you're using static IP or DHCP. You might want to restart your PC, and then launch Dolphin again to see if you still get the TAP error. You'll probably also want to make sure that you can still use the Internet before you mess with Dolphin. If you can't, then you did something wrong, so you should fix it first.
Once it's installed, you should have a second Local Area Connection in your Network Connections portion of the Control Panel, so click and drag to highlight both LAN adapters, right click, then bridge them again. You'll lose the ability to configure the settings on your regular LAN adapter, so you'll need to right click on the bridge to set up your network correctly again from there. How you set up the BBA within Dolphin also depends upon whether you're using static IP or DHCP. You might want to restart your PC, and then launch Dolphin again to see if you still get the TAP error. You'll probably also want to make sure that you can still use the Internet before you mess with Dolphin. If you can't, then you did something wrong, so you should fix it first.