I know some fabrics and thread used to kick up my asthma. I cannot imagine the discomfort with skin allergies with fabric.
Do you have a really nice quilting/sewing buddy nearby. Maybe that person could machine zigzag the edges of the material for you. If not, would you be able to do the edges yourself wearing nonlatex gloves or would that otherwise bother you?
I agree with using the mesh bags [or making some] for little pieces when washing.