It seems that it would be fair for the NPS to let everyone be a TL on one application and a PATL on one other. It would not change anyone's overall chances of winning, would avoid people gaming the system, and every trip could have a PATL. Of course winning a permit would automatically void the PATL status on the other. Even if someone did game the system and had a ringer with high points, the lower points of the gamer would void them out.
For the SJ, there is a note on their page saying a PATL can be added after a permit is won by calling the office.
Vaya Con Rios!