Back in FM10-12 when I was organizing online tournaments we had a snake draft for players. Depending of the number of people who are participating, I would "give instructions" as in "today we are picking a defender and a midfielder". So you can pick whoever you want from those two positions. Tomorrow it would be "striker and goalkeeper" until you have 20 players. You can speed up this process by saying pick 3 players. To keep it more balanced, you can make a list of players available to choose from.
Once that all done, I would load up a editor and create database with all those players in clubs in one league ( haven't done something like this in editor for a while, dunno if its still possible. Ofcourse, people would choose what club they want. After that editing, I would create a new save with only that league loaded and than edit ( with in game editor) all the players/tactics to be perfect ( perfect moral, fitness, tactical familiarity etc) and uploaded it on mach day. Someone would need to host a game, other would join and they would setup a friendly game ( or you can do that via editor if I'm not mistaken) and play one home and one away game. Winner progresses and has a right to swap one of his players out for a player from losing team.
In our LAN parties back in CM days, we used to set up friendly tournaments before a league starts. We used to give each other about a month or so to do some transfers then play the tournament. However, we were 4 of us which made it easier to choose teams that are strong and have budget available (Chelsea having a transfer ban until Jan 2020).
I do not think this method would work (fairly) through forums as with so many players when you are not allowed to choose the same team (we aint playing kid's FIFA here ). There will not be balance between teams if you allowed transfers,
My suggestion would be make a list of teams that can compete with each other, no transfers allowed, maybe giving some time for tactics training as otherwise it is pretty much who's lucky enough to have a squad that adapts faster or lucky.