I like the idea of encouraging to only keep one. Cheap plush is cheap. If you find that you're going through an absolute ton of plush, maybe try and strike a deal with a local thrift store. Or if you want to get real fancy you could do like a "ticket" thing. Getting this animal gets you treat XYZ. We have several locations where we put a colored zip tie on an animal, and there will be an additional prize for getting that animal. Something like a free pizza, pitcher of beer, etc. Of course all of this involves us making money so the scenario is a little different.
Another option if plush is expensive you could go to a mix format. Those nubbly plastic balls are like 25c each wholesale and are easy to grab. We have used those to offset more expensive plush in some mixes. Another alternative would be like 4" capsules stuffed with whatever crap from the dollar store. As long as you have minions to help stuff them, not so bad. You could even put a bucket on the side of the crane to "recycle" the capsules so you can keep re-using them.
Side note, we have started deploying cranes on our route which are filled with empty cans stuffed with expanding foam. When the can is won, they can redeem it at the bar for a "free" white claw or whatever the can is. For an adults party, this could be a fun take on the "bad beer bash" or some other drinking game.
