I tried adding a new route with Dst. Address of my web site, and the second bridge as the gateway.
Am I missing something? Perhaps I should be adding some kind of masquerade, and/or the route gateway should be the Wlan, not the Bridge?
You can only specify an interface as route's gateway if the interface is a point-to-point one. Otherwise, you have to specify an IP address as route's gateway. The latter is definitely the case for the wireless interface.
As your two WAN interfaces are connected to different providers, each of them has a different IP subnet and a different gateway IP in that subnet. To learn the gateway for the WAN interface, configure the DHCP client to accept the default gateway but give it a distance of 2 and more, so that it wouldn't compete with the default gateway configured or received via DHCP from the primary WAN uplink (which likely has distance=1)
Then, display the table of IP routes, you will see the IP address of the gateway received from the wireless ISP. Add a route towards your web server with that IP address as a gateway.
To make it work not only for the Mikrotik itself but also for devices connected to its LAN, your ip firewall nat rules must contain a masquerade rule in srcnat chain for both WAN interfaces (you can use interface list for that).
For a backup or load balancing configuration, you have to configure the routes statically, or use a dhcp client script to change the configuration if eventually you would get a different gateway with each DHCP address allocation (which is not very likely, though).
Instead of writing novels, post /export hide-sensitive. Use find&replace in your favourite text editor to systematically replace all occurrences of each public IP address potentially identifying you by a distinctive pattern such as my.public.ip.1.