Can you paint over or clean Hunter Douglas blinds? cat fur/dog slober

Try to clean them. https://cdn2.hunterdouglas.com/static/documents/Careandcleaning.pdf

1st I would take them down one at a time then vacuum them well to get rid of the cat hair and dust. i am not sure what the material is but if water won't harm them mix a dawn borrax and I prefer ammonia but vinegar is a good substitute with warm water in the bath tub. Let soak for an hour gently hand wash drain tub refil with plain water repeat until well rinsed. To dry take command hooks fasten to bath wall and hang till dry. Have a fan blowing to quicken the drying. If you can't soak them vacuum and wipe with damp rag

Clean them.

There is fabric paint made that you could try. I've painted lampshades and they turned out nicely. I went from a tan to a white on them. I bought the paint at Hobby Lobby but I think Michael's carries it too.

I have soaked blinds and colored Roaman shades in the bath tub with a little soap for a few hours. I used warm water. Scrub spots gentley if needed with a no lint cloth or sponge. Shower spray to rinse and allow to drip out most of the water. Then pat with a towel to dry do not pull or twist them! Then rehang them to finish drying. Hang damp not dripping. This does take time and is a little messy but it has worked for me many times.