Cold weather grooming

Hi horse lovers! (LOL intended)

This big handsome guy is absolutely covered in mud. He's been largely unhandled for many years while his owner was ill and has now passed. I interact with him at the rescue i volunteer for and have applied to adopt him. I took him for a walk (his first time in so long!) Last weekend and did some brushing and grooming. It was about 45 degrees. I want to continue grooming and work on bathing him but the weather where I am is back down in the 20s this week. There isn't a bath stall or even crossties where he's at. I tied him to the hitching post and I'm not sure he's stood tied in years. So I want to be as effective as possible in whatever time I have with him.

He's very alert and sensitive. He is not shut down. He is curious and watchful. He was nervous about any sponge bathing, even just warm water. but he let me brush him for an extended period. Hes got a big long winter coat right now. He doesn't have any health problems, the vet was out and examined him last week. He got good scritches and released after a while.

I want to work up to giving him a full bath over time when its warmer. I have detangler for his mane and tail. What soap is super effective without being harsh? I was looking at the cowboy magic rosewater shampoo and eZall green. I like the ezall green says spray on and rinse. I don't want to overwhelm him with extended bathing time.

What wash products would be good for this situation? Anything that won't be a problem for cold weather and/or doesn't require water? I know he'll get dirty again but he hasn't had a bath in a long time. 2nd pic is what he looks like cleanish. He's a rocky mountain horse, silver dapple.

Thanks!