This is probably a connection through AMS or CDG. Check who really operates the flights AMS-OPO/CDG-OPO and FAO-AMS/FAO-CDG. My guess it that they're not mainline KLM/AF service but some affiliate carrier that is a codeshare partner but not a DL partner and not a SkyTeam member. Hence, you can't book them with DL miles.

My guess is that you'll want a rental car for Portugal (outside of Lisbon), so I would suggest just flying in and out of LIS. The expressways north and south to Porto and Faro respectively are very good and not particularly crowded as they're toll roads. [Rental cars have transponders and you're automatically charged months later, so you don't have to worry about cash/credit cards or waiting at toll booths.] Much of the way is a very pretty drive. However, when I did this (late June) there were extremely long (hours long for some people) lines at the LIS airport rental car counters. Try to use Hertz Gold service or something similar if at all possible.

Note however that the DL TATL JFK-LIS is operated by a 757 with D1 (IIRC 16 seats) and is seasonal, so you might prefer to do a connection through AMS or CDG to LIS, which IIRC was an AF Joon route last summer but Joon is gone now.