Cutting white fabric

You are not the first one. There was talk already.
Once you get more power to cut white - new problem will be charred edges.

If you are willing to experiment - I can suggest something: this is fundamental physics problem: white does not absorb light, not enough heat dissipated to cut/burn/evaporate the fabric. Need to increase light absorption. It should be possible to cover white fabric with light absorbent, like dark water paint, dry it then cut then wash. Water paint should wash off without any problem. Wet will not cut as water will absorb most energy, must be dry.
You may still have problems with charred edges. For that the idea I have is to use air assist with non-flamable gas instead of plain air. Inert gas is the best, cheap CO2 will also do.
You have blue laser, it might be possible to use CO2/IR and use IR absorbent die that is visually white. That would be different machine that is more expensive than LED laser.