It would be best to take some clean plates of the background and mask out the rig. The Wire Removal tool is only good for very small things like, wires/monofilament -- because it works by mirroring the image nearest the point and blending the two halves, it gives an effect similar to the smudge tool in Photoshop -- So, if your rigs are any thicker than a wire (and I'm betting they probably are) then you should definitely mask them out and add your clean plate.
I do my wire removal in TV Paint Animation, but I think the process would be similar in After Effects. You load the clean background image in on one layer, then the animation footage on another layer above that. Then you can erase the support wire, revealing the background image underneath. (Or use a mask, only I don't know how to do that, I don't use AE for much besides greenscreen keying.) If you didn't take a background image without the puppet and rig, sometimes another frame, maybe 3 or 4 frames earlier or later, will have the rig in a different place, so you can erase through to the background in that.
The main thing to watch out for is if the puppet casts a shadow onto the background behind the rig, so that when you erase it, the clean background without the shadow will look different.
Also remember if you're moving the character out of your focal point (toward or away from the camera) you'll need to take plate shot of the newly focused position. Especially needed with a lower FOV, but something to keep in mind.
I erase rigs in TV Paint, which isn't much help to you if you are going to use After Effects. But both programs do load layers, which is what you need.
Here's how this one worked:
1. The first frame I shot, the raven had not yet flown in the window, so I could use that as my clean background plate.
2. I loaded just that one frame, as a still image. I could then drag that layer out to cover as many frames as I needed. Here there are 50 frames ( 2 seconds at 25 fps).
3. I loaded the whole sequence in, as another layer in the same project. It appears on the top layer.
4. I go to the first frame where the rig is showing and paint out the rig by setting the tool to "erase" instead of "colour". I do each frame, one at a time - in this image I am up to frame 17. I am using the hard-edged Pen tool here, but I could also do it with the soft edged Airbrush tool to the left of it. As I erase it, you can see the layer underneath, which has the same background but no raven and no rig. I carefully erase around the raven with a smaller brush size where the rig touches the bird. I also had to watch out for the shadow of the rig on the floor, and paint that out too, but leave the shadow of the bird.
After Effects and Photoshop have an erase tool, instead of turning the paint brush tools into erasers, but apart from that you could do something similar. There is also a masking tool in AE that I don't know how to use, I think it can be animated to follow the rig across the screen, then you just erase what is inside the mask area... or outside it... not sure because I don't do it that way. But there will be tutorials on Youtube, there are for most things in AE. This is how it looks in TV Paint Animation. You will have to click on the image to enlarge it if you want to read the text:
If I had shot the raven against a greenscreen, I would have just painted the rig out with green colour, just on the one layer. Then I would import it into AE for keying out the green. In this case I wanted the raven to cast a shadow on the set, so it felt like it was really there in the room. And in the next shot it had to land on a bust above the door, so it needed to be there. But for a shot with it flying outside, I used greenscreen, painted out the rig, and keyed it over the exterior shot of the sky and house which were in a different scale anyway.