Phrase Choice – ‘Should Probably Not Check’ vs ‘Probably Should Not Check’

In this context, I would say the noun form of "authorized" would be "authorization," not "authority." In other words, "No authority" would be better expressed as "No authorization".

The words have multiple meanings, some of which overlap a bit. Generally speaking, though, "authority" means you have the power to do something, while "authorization" means you have permission to do something. One could argue that credentials grant you authorization, but not necessarily authority.

As someone else has already said, there is more than one way to convey this. Another you might consider is:

• Insufficient Authorization

1) To me as a native BrE speaker all of those sentences are OK (apart from correcting "wonder" to "wondering"). Although "check out" originated as AmE it is now heard in BrE too in this context and others. I don't see much difference in the degree of formality. In my experience tradesmen and women are not much interested in formality as distinct from politeness. All the sentences are quite polite with the third being very much so.

If you were going to take a vote, I personally would not use check out, but I would use the third sentence.

2) Other terms: "house call" is associated, but not exclusively, with visits at your home by your doctor. A quick duckduckgo search reveals that there are lots of companies offering PC repairs "at home". Another term often used is "call out": sometimes you are offered service at home with no call out charge, other times you have to pay such a charge. In my experience different traders use their own different terms.