Ground rules:
1. The "Big Six" franchises at DC here are the Superverse, Batverse, Wonderverse, Flashverse, Lanternverse, and Aquaverse.
2. "Franchise characters" here extends from their respective Main Alpha to their primary love interests, sidekicks, supporting characters, and intentional knockoffs (i.e. Red Super-Man)
3. You can't use ANY of the Big Six alphas nor the rest of their supporting cast for your JL roster (this means all current/former Robins, Batgirls, Superboys, Supergirls, Wonder Girls, Kid Flashes, Aqualads, Aqualasses, and non-main Lanterns are off the table, as are Mera, Catwoman, Steve Trevor, Lois Lane, Carol Ferris, Iris West, etc).
4. Big Six franchise characters are off-limits, but non-Big Six franchise characters normally associated with other teams are definitely not (i.e. Titans, LoSH, Doom Patrol)
4. Minimum roster is 7 full-timers, maximum is 13 members but split into 2 teams of 6 with one 1 overall head.
My first crack at this, here we go:
Minimum Roster
Martian Manhunter - mentalist
Black Adam - powerhouse
Vixen - versatility
Cyborg - technologist
Zatanna - magic-user
Hawkwoman - martial artistry
Plastic Man - humor