Green Screen Team Photography — is there a shortcut to this?

[JohnFoxtrotS]

Hi, first post. This seemed like the appropriate place to ask this question but if not sorry.

In the fall my business is almost all team photography and, being in Texas, that generally means we get the business of football teams, then the rest usually comes with it. Between my company and my dad's company we photograph about 70-80 colleges, high schools, and middle schools.

Our main draw is the custom designed risers we have that help us create the best height and spacing to see every player and their jersey number clearly.

Currently with school closings and the future of fall workouts and practice times in limbo I'm looking at what other solutions we can offer for this. 70 team photos is a lot (and that's 70 schools, so realistically we're talking about close 200 actual team photos), and each team photo has anywhere from 40 to 100+ athletes in them.
Obviously green screen and building a composite image of the team is the solution, but time is the problem. Often we need to turn-around 3 team photos (Varsity, JV, and Freshmen) as well as varsity headshots by the next day for program publishing.

Is there a green screen application (or if you have another solution in mind...) that allows you to quickly insert several images into a photo and arrange them as you might arrange layers in Photoshop? I'm including a sample of one of our traditional team photos to show you the sort of results I'm after. It would be great if I could produce something like this with no more than 2-3 hours of post production.

Thanks for your feedback.