The thing to do is to exploit the cats' natural hygienic nature. Yes. I know that sounds like a contradiction in terms, given the circumstances, but bear with me.
Cats will not foul where they eat. If the missing cat turns up in your hay, start feeding it - in the hay. Leave a bowl of dry food out and move it every few days all around your hay stack. As a bonus, the cat will keep down the rats and mice that you will have in your hay (and you will have them, I'm afraid). The more the cat feeds there, the less it will foul and the more it will hunt.
Mostly, cats pick up the parasite from eating infected rodents. Mice and rats are notorious for their "nibble and dribble" method of eating, so if anything, they are your real problem. Cat faeces are not contagious for two days after they are deposited, so clean up as soon as you see them at the same time as the feeding programme. That way, the cysts will not have had time to develop and become infectious.
This is incredibly concerning for you, I can tell. When you have a small child, toxoplasmosis is a real and present worry. It isn't something that is tested for in rescue centres, for the simple reason that it is a parasite that occurs in most animal's faeces, including human. Astonishingly, it is estimated that 50% of the world's population carry this parasite.
This isn't the cats' fault, honestly. I can understand your frustration and worries, but try and see it from the cat's POV. There is a cold and wet garden or field, or nice, warm and dry place to go to the loo. Which would you choose? Neither they nor their owners are doing this deliberately to annoy you or put your family in danger. It is more a case of impotence, I think, on the part of the owners, and nature on the part of the cats. I'm not trying to downplay your hair-tearing frustration, anger or concern, really I'm not. What I'm trying to do is give you a solution where everyone wins.
It is very difficult to stop cats wandering around. If they are going to come onto your property, and it is almost impossible to stop them, make use of them, work with them. It is worth a couple of boxes of cat food, surely?
Best of luck.
