We are living this at my house - have 2 cats and wash their rugs in the washer. I did the garden hose with same results you got. Damn washer soap combined with hair just slows the drain and the washer pumps normal overloading the drain.
We have helped the drain by pouring commercial draino (any major brand) gel solution down the washer drain; it settles in the Ptrap and then running a hot water cycle. Container says clear drains in 15 minutes but I let it sit overnight. Still not out of the woods, but getting better. I also shoved the drain hose over a foot down the drain.
Neighbor with a cat shared this fix with us. Good Luck.