You are right. Taking all the dogs away is going to be traumatic to her. They have become her comfortable obsession that allows her to deal with the chaos that is in her mind. She relates to the dogs. The rest of you humans are weird while she and her dogs are ok. CLASSIC! The fact that money was disappearing to dog charities is not a surprise. She has no idea that she has done anything wrong by sending money to these dog charities. They need her help and she feels control and satisfaction in her ability to help. Yet you can not let this situation continue so the question is... what to do!? The problem is there is no good answer.
Your best solution is to deal with each dog and not try to get rid of them all at once. You are probably going to have to be a little deceptive as well. The idea of full disclosure and total honest is out the door as of now. If you can find a reason that one of the dogs needs to go to the vet then use it to your advantage. Take the dog yourself. Find a reason the dog needs to stay gone and then turn it over to somebody that will find it a good home. Let that settle down while controlling the intake of any other dog. Try to keep her away from information regarding neglected dogs. Block sights from the internet if she is on the computer. Control the money flow so she can not buy another rescue. Then you can do it again if necessary to get rid of another dog. But leave her with a few dogs because they are her comfort blanket.
My Mom's obsession was pine straw. 103 degrees at home and Mom is in the flower beds picking sticks and leaves out of the pine straw. She would fluff it up and put down hand fulls of new straw where needed. She did this for weeks!! It mattered not the weather. She sat out there in the blistering heat and the rain. I made sure she ate (sitting in the flower beds many days), had plenty of hydration, and would actually stand over her with an umbrella to shade her. This went on for frequently for 8 months!! Her other obsession was washing windows. Sometimes the 24 outside windows were washed twice a week. This involved ladders. I tried to to show her the leaves in the straw when window obsession struck!
When you get your diagnosis please talk to the neurologist about your wife's dog obsession. There is medication that will help! Medication that does not sedate but will take the edge off the anxiety that she is feeling which feeds the obsession. Yep, we are back to diagnosis