"He who marches out of step hears another drum." ~ Ken Kesey
putting myself in this spot, I think very first thing I would do - which I'm sure you will too- have them sign some sort of a contract, "not my responsibility if these over prints fail, change color after washing, spontaneously combust, stop saving the whales...whatever." 1) I'd talk them into a shape that would cover without alignment being an issue, like a solid circle or a heavy outline.2)Test every transfer product I can get my hands on. We used to use some EcoSol stuff back in the day that would stick to nylon, but it looked and smelled like a shower curtain. Check with Supacolor, That would be my first stop.3) start shopping for a new Lambo for the price Id be charging to get involved in this mess..
personally i'd hire a very high end ink / printing consultant. Like Coudray, Grieves, or even Danny to figure out how to screen print this. There are tons of nylon inks, waterbased or plastisol that will allow you to print these. Even if you have to buy specialty pallets.Transfers are the quickest solution, but if you only have 3 heat presses, i'd invest in more of them.