I have done many with the bushing cloth you have. Unfortunately the holes are not all the same in the flanges so pre cut cloth is useless. Especially since they cut them all the same width. I had some that the cloth was not wide enough so I had to cut the pre cut off to start one strip. Now I have a bunch of short strips that I use to prop up the hammer rest rail when it needs it.
Make sure the hole is clean. Many times they didn't use any glue, or it seems so. You can resize the hole with a drill bit but you really just want to clean it. Hide glue is hard to see but adds a lot in the hole.
If it is pulling too hard, yes you can grab a thread and pull to thin the cloth. Works quite well.
However,,, take your larger needle nose pliers. feed the pointed end through the first hole. Grab the strip with the pliers and rotate the pliers, leverage against the flange pulls it through just enough to get the strip through the next hole. Grabbing the cloth tight to the flange and rotating the pliers, you can pull the strip through to the end. Hold the other side of the flange as you pull so it keeps the pressure from snapping the unsupported wing of the flange.
Then with a toothpick or needle, place a thin line of cold hide glue or what ever is compatible with the old glue,, around the cloth and pull in to the hole,,, both sides.
I then have a #18 pin that I push through to align the cloth and compress. Set it aside until the glue hardens.
Slice with a good blade and slowly ream with the Mannino reamers to the size pin needed. Sometime the hole needs to be sized with some 70% isopropyl alcohol and then the size pin can be smaller.
The more cloth you can get in the hole,, the firmer the bushing. Check the side to side play to see.