There is a point that often gets missed on this forum, so I'm going to explain it. This isn't directed to you, but if it works as a reminder, that's great
As soon as you put IP addresses on bridge1 and bridge2 from my example above, those L2 domains can now talk to each other over L3 (IP). But they remain separate L2 domains.
So if you put 192.168.1.1/24 on bridge1, and 192.168.2.1/24 on bridge2, suddenly all devices attached to each bridge can now ping each other.
I see a lot of posts that assume something special needs to be done to get those networks to talk to each other, or to talk to the internet for that matter. If you're using the default masquerade rule, they will both talk to the internet. If you have no firewall rules specifically blocking communication between those 2 bridges' subnets (aka L2 domains), they will talk to each other because they are connected routes.