Ditch the bridge. You don’t need it. I’m thinking of two ways you can take this on.
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This if your HA device has 1 ethernet port. Get a cheap smart network switch, like a TP-Link TL-SG108E. Use nmcli and create a secondary virtual nic within HA. You can segmate the networks with vlans and use the switch to pass traffic between both, allowing HA to both networks. More info about nmcli here - Configuring VLAN adapters on Home Assistant Yellow - #5 by justynnuff
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Get a 2nd HA device. You can get this done relatively cheap. Pick up something like a Dell Wyse 5010 on eBay for $10-$20. You can keep the networks completely seperated and there are ways to link multiple HA instances.
[EDIT] FWIW, I use the first method. I use OpenWRT and have a main LAN network and an IoT network. I had to use nmcli because I only allow media players on my LAN network, everything else is on the IoT network, and mDNS and AVAHI are a pain in the butt when it comes to discovery, despite allowing firewall access from my LAN to my IoT network. Nmcli allowed HA to sit on both networks and solved my mDNS and AVAHI discovery issues.
For the network gurus, I did try reflectors, but I couldn’t get it to work. Nmcli was much easier to setup.