Actually they don’t need to trap them. They already dart a lot of deer with a vaccine to a pregnancy protein, PZP, as birth control to limit the growth of populations in some areas.
I have no idea if this is true, but if they want that mRNA in game meat, it would be feasible at some level with darting, and limited only by how much they were willing to spend.
As for the keeping it a secret, you have no idea the big secret being kept from you right this very moment. Keeping secrets is no problem for the intel op running the country right now.
Even darting 100 million sounds preposterous.
Wait a sec...I can see that many right outside my kitchen window. We are overrun by the damn beasts.