As "dbfilter = ^%d$" is automatically cutting off "www." it works the same way if you define a virtual host for each domain name, such as one virtual host with ServerName www.website1.com and another one with ServerName www.website2.com, means www.website1.com becomes database website1 and www.website2.com becomes database website2.

This is my Apache configuration (sub1 is the name of the database). What you are looking for is the X-ODOO_DBFILTER variable. dbfilter will pick this up. Adapted for your example:

<VirtualHost *:80>
    ServerName www.website1.com
    ErrorLog /var/log/httpd/www.website1.com-error.log
    CustomLog /var/log/httpd/www.website1.com-access.log combined
    ProxyRequests Off
    Header add X-ODOO_DBFILTER "website1"
    RequestHeader add X-ODOO_DBFILTER "website1"
    LogLevel warn
    <Location /web/database>
        Order Deny,Allow
        Deny from all
        Allow from my.home.ip.address
    </Location>
    ProxyPass /server-info !
    ProxyPass /server-status !
    ProxyPass / http://localhost:8069/
    ProxyPassReverse / http://localhost:8069/
    ProxyVia On
    ProxyTimeout 300
</VirtualHost>

And you can repeat this section for every new domain.