I am getting very close to finishing my set and I am planning for filming. I'm somewhat nervous to animate my characters because I'm not using tie-downs. It was a choice I decided not to do since I did not know enough about it and it seemed like plenty of work (and I have a limited amount of time). My characters for animation are fairly tall (range from 8 - 12 inches) and quite mild. I'm thinking of using sticky putty at the bottoms of their feet but I know that's not going to be perfect.
One alternative to using tie-downs is a rig, but that will involve you in painting it out in post, which is a complication. Unless you can hide the rig behind the puppet.
Another technique is to use magnets underneath the set. For this you need a floor made of thin perforated metal, and you need strong rare earth magnets.
Perhaps the simplest would be to make the set floor out of a solid foam, then use pins to fix the feet to the floor.
I would say, use tiedowns of some kind if at all possible. You will spend more time animating without them, than you would spend making them.
What is in the feet?
Sometimes I just put a screw down through the foot block into the particle board set floor - usually if the puppet has gone too far to the back of the set and I can't reach underneath to grab the tiedown and above to hold the puppet at the same time. The edge of the set floor digging into my neck puts a limit on how far away I can use the regular tiedowns, but sometimes I need the puppet to take just one more step further away. But I usually have a metal block in the foot with a hole in it, so there is something solid there.
If you have just a wire loop in each foot, that may still give you something to put a screw through. Simple tiedowns can beamed with a nut glued on top of a loop of wire, with a little epoxy putty around it to hold it in place. Thin half-nuts or square nuts might fit better than the regular thick nuts. Or you can file a groove around the outside of a thick nut, and twist the wire around that, so you don't get the thickness of both the wire and the nut.
Even with a rig to hold the weight, it helps to have something, a pin maybe, that keeps the foot in place so it doesn't slide around while it is supposed to be taking the weight. Sticky wax or Blu-tack might work for that.