There is no requirement for ClearCenter to be the domain registrar but you will have to use subdomains. Does company.com resolve to your website or is it just www.company.com which resolves to your website? It is not that important.
If ClearOS is on a fixed IP, you can set up an A record for subdomain.company.com, which I assume will be industry.company.com, pointing to your WAN IP and your Internet Hostname should be set to subdomain.company.com.
If ClearOS is on a dynamic IP, you can set up a CNAME record for subdomain.company.com pointing to your poweredbyclear.com dynamic DNS name. This way subdomain.company.com will always follow your IP.
If you do either of the above, you can even run a web server on subdomain.company.com but it just has to use a different subdomain from your remote server.
You don't need any form of DNS registration at ClearCenter set up to subscribe to ClearCenter services as ClearOS looks after contacting the ClearCenter servers.
I see you mention Content Filter. These days it is only really effective if you use a non-transparent proxy as the transparent proxy only filters http traffic and not https. This is relatively intrusive as it means making proxy settings on every PC, or setting up Web Proxy Auto Discovery (WPAD). The app needs some love from a volunteer but there is a HowTo to do it manually and it is not too difficult to implement if you know how to write WPAD files (I don't). As an alternative there is the Gateway Management product which is easier to set up and much lighter on resources and is ClearCenter's preferred solution.