First I suggest the name is much more important than the logo, forcing yourself to only use a name that fits a crown may not get you the most engaging, memorable name.
The words I cone up that are associated with a crown like Royal, King, Queen seem rather old fashioned.
To find what's available, one can use the bulk search tools at domain name registrars like godaddy.com to check web site name availability of a list of name ideas at once,
There are sites with tools that help you come up with fictitious names, including: wordlab.com, naming.net, wordoid.com, lingzini.com, namestation.com, domainsBot.com some of these tools look up the domain name availability as you go.
If you find an available .com name, there's a high probability that it' not a trademarked name.
You can also do a Trademark search at USPTO.gov