Try making a solution of "No Rinse Body Bath", flip that rug over and spray the stained areas. Wait a day, and see if need more. If there is any strange odor still the do solution of No Rinse with food grade or technical grade hydrogen peroxide. Drugstore version might have unlisted chemicals that might affect this expensive rug. i have used this mix for years with execellent outcome
I would pour plain vinegar on the spot. It should not harm the rug and the rug has already gotten wet. When I do this I soak the area with vinegar, let sit for a few minutes, then step on the spot with a towel to remove some of the dampness. The vinegar smell with disapper in a day or two. I've also done this on sofa cushions and they have been fine.
I can tell you I know what you are dealing with. We moved into a house that had dogs, cats, and babies all urinating on the carpet. I took the shampooer and put in 1 cup of Tide and 1 cup of bleach the rest hot water and put it in my shampoer. Worked great! Vinegar in a spray bottle works too. FYI on a mattress that has an odor soda or baby powder is good. I used to keep a spray bottle handy.
We advance to the new stuff for oders. Fabreeze although we are in a different house, we all love the nice smell and it does take out pet odors I just remembered my brother uses it!

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