I've never tightened a kaskara hilt, but on "first principles", I'd proceed as follows.
The blade and hilt are held together by two wooden pieces that forms the basis of the grip all wedged together. Some blade tangs have a hole in it and the wood and blade are joined by a pin through the wood pieces and the blade tang that also secures the hilt. The leather grip wrap could be removed and the assembly "structure" could be assessed. Even though the grip wrap appears to be "modern" and not of the period of the hilt and blade, I wouldn't recommend going this way at first.
You mentioned a piece of cloth stuffed up into the area between the blade and the hilt "struts". I believe this is a hilt tightening technique. (I have two kaskaras with what looks like thin leather stuffed into this space. Both have tight hilts. Also, I have another that is tight as well, but without any stuffing.)
If the stuffing is leather, a few drips on leather preservation oil might replace the original moisture content and make it a tighter fit. If that doesn't work, I would remove and repack the stuffing to see if it does the trick. Can't hurt. As a last resort and if I was confident I could properly re-wrap the grip, I'd remove the wrap and investigate the guts of the assembly.
Let us know how any of these approaches work.
Ed
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