Why not tertius cliens?
No particular reason. Both word orders are correct. Bringing tertius before cliens adds emphasis to the number.
Why not "customers" here?
I know that in the shopping module, "emptor" is "customer" and "cliens" is literally where the word "client" comes from, but I still believe the two are synonymous.
No, an emptor is a purchaser or buyer, or one who obtains by bribery. A cliens is mainly a person who attaches himself to one of greater influence or political power (a patronus) for protection.
Again, they correct my mistakes in English. But I am not studing English ☹️
why is it not the third patron
Tertius is nominative case, agreeing with cliens. If it were the third client it would have to be the accusative tertium, agreeing with patronum.