The only clients I had that had a mastitis problem were milking the mare.
Found out the mare got mastisis every year....
I flat out told them I never milked a mare and I have never had a problem. They stopped and have not had a problem after (been 10 year)
So dont touch and dont worry about it.
Once when I wean the mares do not go back with the weanlings. I feed weanlings and yearling different then the mares.
I have seen mares (even barren ones) start to produce milk when ever they are sucked on. So at any time is the young one tries to suck and the mare allows it milk can be produced.