What utter bureaucratic stupidity. Using the LAMP tests meet the requirements that the government has published (ie the timeframe and the type of test).
So by their logic, doing a self-administered / proctored LAMP test - but purchased from an American company with an American address showing on the PDF - would qualify?!? Even though it would probably also be the very same Lucira test kit? Ridiculous.
The point is to show that you don’t have COVID, using the required testing method, within the required timeframe. Period.
You did exactly the right thing. By the book. If they’d given you the threatened penalty, I can’t see how any court would side with a bureaucrat on a power trip.