Coming from a country with excellent universal healthcare, I find it absurd that you should need to "shop around" for any procedure.
For this case, it is silly for a CT. However, for procedures and surgeries, not all doctors are equal. Do you have any choice in your country?
Yes, you can absolutely choose. There's a web site for finding hospitals/clinics for your particular procedure, although your doctor can also refer you directly. The web site tracks the wait time and (often) publicly available quality metrics collected by the health authorities. Travel expenses are covered to some extent if you don't want to use your local options.