You do need the module dbfilter_from_header in order to set up multi website theme properly. My configuration:

- odoo.conf: disable dbfilter, add dbfilter_from_header as a serverwide module

- proxy virtual host definition (in my example it is my caddyfile for the Caddy Webserver, but you can easily adapt it by adding the request header X-Odoo-dbfilter for your use case):

mydomain1.ch, www.mydomain1.ch {
  proxy / http://192.168.xxx.xxx:8069 { # Fill in the correct port..
    transparent
    header_upstream X-Odoo-dbfilter mydatabase
    }

  gzip
  log /var/log/caddy/mydomain1.access.log
  errors /var/log/caddy/mydomain1.error.log

}
 
mydomain2.ch, www.mydomain2.ch {
  proxy / http://192.168.xxx.xxx:8069 { # Fill in the correct port..
    transparent
    header_upstream X-Odoo-dbfilter mydatabase
    }

  gzip
  log /var/log/caddy/mydomain2.access.log
  errors /var/log/caddy/mydomain2.error.log

}