My personal opinion (reading the info provided by IATA) is that doing MIA LIS on TP followed by LIS IEV on W6 cannot possibly work for him - having the test will not save him (this is required because he is arriving from USA), he will not be allowed to pass through immigration and will have to pay out of his own pocket to fly back. I still think that he would (should) be denied boarding by TP in MIA, even with the test - however, if the local staff does let him board and then he has to do an immediate turnaround once in LIS, I would not rely on the mercy of TP to make the arrangements for him free of cost (as customary, per CoC standard language, it is entirely and only his responsibility to ensure that he has the required documentation to enable him to legally move throughout his trip).
As you suggested, the only viable out-of-Schengen transfer possibility I see is via IST on TK.