So, for the avoidance of all doubt, I booked the flights on the Lufthansa app. It clearly says Economy Basic, and by each of our names, it says we have 1 checked bag. Both can be seen in this screenshot from MMB:


I have added my United FF number to my name for any *A benefits I might be entitled to. If I do need an extra bag, I’ll just rock up at the airport and show them the information on the *A website, if there are any objections.