There is no reason to soak the snake at all.
I've treated numerous collections and have not needed to soak a snake to rid the collections of mites.
The treated paper towel and the treated tub is all that's necessary to treat, the bugs will literally fall off the snake over the first few days after the initial treatment.
After the 2nd treatment the rest fall off and die and by the 3rd treatment there is no live bugs.
There's no need or reason to use soaps or oils or anything other than permethrin on the tub and the paper towels.