Hosts generally let you add any domain, then provide you all the DNS records you need that you can add into your Cloudflare account. Is your host not letting you do that?
They may not have competent people to help you, but they are a genric cPanel hosting provider, and nothing should stop you from doing this.
Just log in to your cPanel account and obtain the IP address of the server you’re on. Then go to your Cloudflare DNS page and create A record(s) pointed at the IP address you picked from your cPanel dashboard.
NB: If you want to use the cPanel-based email that comes with your hosting, create a separate A record for a subdomain (eg mail.britoiloffshores.com) going to the same cPanel IP address. Keep this subdomain record DNS Only and use it as your MX record.