Somehow I don't think that's where he got the idea, at least not directly.

We see that Strzok was researching it the day before Trash Heap Joe says "Logan Act" in the famous White House meeting, and that he and Page were conversing about it.

Remember that The West Wing was not just pure fiction - it was influenced by the personalities and events of the Clinton Administration in particular. Dee Dee Myers was an important consultant to the writers. Maybe Logan Act spying actually happened at the end of the Clinton era, or even if not, it might have been seriously considered. Anyway, I think that's where the idea came from; I don't believe that the Hollywood writers just pulled 'Logan Act' out of their asses.

I have two speculative theories:

1. Valerie Jarrett and Ben Rhodes (a fiction writer) are trying to figure out how to burn Flynn, and Ben remembers the West Wing episode. Somehow they get that idea to Strzok and Page, and to Trash Heap.

2. A Clinton admin alum, involved in the 0bama admin, remembers the idea from the old days. This alum must get the idea to Strzok/Page and to Trash Heap. The West Wing episode is not involved. People in the chain could include Bill, Hill, Tarmac Loretta, Susan Rice ,,.