I'd give it a go if you have some extra time. As I was writing you by PM, I haven't personally heard about people just rocking up and getting a test done, but I've read of people doing it (here in this forum).

Have you thought about the Auxologico? From what I heard it's a bit more reliable than the San Donato. BTW, if I have a moment today I'll try and give them a call.

While I'm on the subject, a quick word about the 'codice fiscale'. People, especially in the US, think it's like their SSN - it isn't. It's just a code that is made with parts of your surname, DOB, place and province of birth (can be abroad of course) which you need to use to pay taxes and access public health. You can generate one from websites and it's totally OK to self create one. When my now wife moved to Italy to study we went together to get it done at the local branch of the agency for revenue collection and she got one before I could complain about how inefficient they were, which was a surprise.