How to enable CLR on a remote SQL Server?
SQLCLR, or technically the "CLR enabled" server configuration option, is a Server-level option, not Database-level. Since it sounds like you are using shared hosting with no server-level permissions, then the answer is just what the sysadmins told you: No.

Now, if you can convince them to enable it for everyone on the server, then you would be able to do this.

Or, you could possibly consider using SQL Server on Amazon RDS which allows for SQLCLR (SAFE Assemblies only -- I have successfully installed and used SQL# on Amazon RDS) and should have options that are not cost-prohibitive (though I don't know your specific requirements, but worth taking a quick look as they do have a "free" tier).