I can't make out if you are planning to cut the whole tail or a single piece underneath that is coiled and dirty?
Under no circumstances would I support removing the whole tail back to the dock - there are still loads of flies and biting insects about and dirty or not a tail is the only protection for the back half of the body, particularly the udder is at risk of infection if there is nothing to swish.. And then when the bad weather comes she'll have no protection to her hind end which is what is turned to the wind and rain because it is best protected..
If you have washed and rewashed the tail I am not sure why you can't get the dung out - it takes more than a couple of shampoos and most equine shampoos are for showing and focus on shine rather than dungpower
I'd recommend fairy liquid neat on your hands to rub the dungy areas til they're soft and can be picked out. Then a tub of soft soap (Clydesdale guys swear by this stuff!) from your local agri merchant. By all means some hair may have to come away but not all of it. And after every wash run pig oil through the tail as this helps coat it against further dunging and keep detangled for tail rubbers.. I have washed the tail of an elderly mare who had a bout of bad scours that was left unwashed for a few days because the owner had young puppies in the house and was wary of taking the chlostridial infection indoors, it took me 3 hours but the job was done.. With serious cases such as you describe it may take 2-3 sessions over a week of 3 hours each time but pig oil in between will help preserve the progress and you will get there
I'd also look at the causes - yes it's a vicious circle now but did/does she have a GI problem which led to scouring and started this off? Does she have arthritis or any other form of bone damage that might result in her not being able to lift the tail sufficiently? In which case maybe she needs supplementation to resolve the problem so it doesn't recur so badly..
I remember a stallion many years ago who had damaged his tailbone during a fall and couldn't lift the tail enough - he started with a dungy tail and started to suffer colicky symptoms as the pain led to less lifting and the dung started compacting.. Not scaremongering but worth checking if she is sore anywhere in the back or back end as a precaution.
I hear what others are saying, I also think it can be done and is worth doing without removing any significant portion of the tail. If you're talking a single matted bit it may be easier but regrowing stubble can cause irritation inside the dock and hind leg soft skinned areas and although it's just my opinion, I'd keep removal as a very last resort in cases where the dock itself is broken or damaged to a point it's unable to lift..
Oh and the way with your advisers on the "no you shouldn't" bench is to rope them in to help you get the tail washed out and sorted
I'd volunteer but you might be a bit far for a daytrip and it's a long ride home with ponydung smell lingering all around you