Do you really need the bath mat?
Sorry if I sound dense... but, what does using a bath mat, have to do with your child's sensitive skin????
Bath mats, are Latex or some other rubber. For some that creates an allergic reaction.
Does your child sit ON the bath mat, while in the tub???
And can't just sit in the tub, itself?
Just use a towel for him/her to sit on.
Anyway, for your bath mats, REALLY rinse it out after each use, very well. Because, any soap foam and slime or whatnot that is still there via residue, will also create, mildew on it.
THEN, HANG UP the bath mat.... like hanging laundry, to dry it completely.
Keeping it outside, the bathtub or enclosure. Because, leaving it in there, it will have perpetual moisture and not completely dry out.
Do not hang it up against the wall... there is no air circulation there and it will still be moist and not dry completely.
Ya gotta hang it OUTSIDE in the air. In the sun.
With freely circulating air on both sides of it.