To print papp resist on fabric using KATA stencil, you can use common silkscreen squeegee. At the Zeugfärberei, they have the wooden spatula that is used in KATA-zome in Japan in some regions. (some also use very thin wooden squeegee).
Place the KATA on the fabric, where you want to print. Remember where papp is on remains white and the rest will be colored. Place papp on the KATA and spread/press it down to the fabric. This is similar to how you apply silkscreen print pigments. I leave then extra material on kata before peeling it off. This allows me to gain thick layer of papp on the fabric. But this is a bit of preference. For example Janina squeeze extra papp out before peeling it off (like how you do in screen printing)
Then peal the KATA off slowly. I lift one end and slowly role it out. This way the screen on KATA releases the papp on the fabric better. If you peel it off too quick, the papp sticks to screen side and comes off from the fabric. You can fine tune your technique how to print papp perfectly on fabric as you practice.
After printing is finished, wash the KATA using cold water. Papp is water based and it comes off easily with water. I used brush to make sure it does not get stuck on the screen or the detailed part of the KATA stencils.
This is a self-made tool I learned from a KATAZOME craftsman in Japan. It helps to keep the KATA from touching the already printed papp when printing near by.