Ami: I can't comment on the 'mix', not knowing what the components are, but can say that it is producing a trait which should not be bred through.There is a limited market for the 'Permanent shed pooch' and , even if it breeds true for generations, I can't see the Kennel Club admitting it to the Register

If ever they did they'd still have to put it into one of their 'groups' for showing.It could hardly be what our KC calls 'utility' as it has no utility, but 'working' is a possibility: it would certainly get its owner working with a brush and broom

You certainly know how to set a question. Sounds like a history test " Compare and contrast a) the wheaten terrier b) the golden retriever c) the labrador d) the standard poodle e) any
two other breeds, giving their roles in and the causes, course and consequences of, the Great Shed Rebellion of the early 21st century" [Candidates may use illustrative graphs.
Write on one side of the paper only]
Right: Off topic. GOD (Grumpy Old Dogman) writes: " If you choose a golden retriever remember it was meant to be a
golden retriever. It is not a platinum retriever, a beige retriever, a silver retriever or a white retriever.Nor is it a monochrome retriever.It should be a dark gold, at least 'old gold', or light mahogany, with very pale cream or near white feathers [the fluffy fringe bits are 'feathers']" Many breeders insist on breeding what are literally pale imitations of the true original.One here excused this line by saying that it was done for the 'American market', a slur against the supposed taste of Americans which was so vile and insulting that I choked on my Pepsi!
Of course the golden moults. Nearly all breeds moult, as you appreciate. Moulting is not meant by nature to be a daily event all year round but we insist on keeping our dogs in centrally heated, constant temperature, inside all year round. So the dog doesn't know what season it is in.The golden retrievers we had here in Cambridgeshire back in the 1940s and '50s moulted to change their old winter coat for a new, summer one
but they only got warm in winter by sneaking into the kitchen to join the rest of us huddling around a smoking labourer. And when they did moult they did it so well that we saved on having carpets for the whole twelvemonth: we had instant deep,soft, wool all over the floor.Rushes on the floor were suddenly
so last Age
Now you have to comb and brush it every day to get the loose hair out, a bit at a time.
It has a wonderful temperament though. (One of ours used to kill the farm kittens but, hey,everyone is entitled to a hobby)It's a gundog so it's obedient, very faithful and sticks pretty close to its favoured owner. Not the brightest dog but consistent and trainable.
Wheaten terrier ? This must be what we call the 'soft-coated wheaten terrier'. Good news for the breed. Somebody wants to keep it going,
and quite right too ! Its numbers have been in decline here in Britain. One thing to point out: It's
Irish 
The phrase for it is 'happy go lucky'. It's extrovert and exuberant.It needs
patience to train (so you'd best find little Patience, the fair colleen, to help you

). It's wonderfully eager to please and loves children (assuming, as always, that the children are taught about handling any dog). The soft, silky, coat is low maintenance. The coat doesn't change from season to season:we don't think of it as a moulter but, of course, it does shed hair in the course of the days. Remember it is a
terrier. Terriers were bred to go off and dig or to kill rats, not do obedience trials or follow some gun's orders. Note:If you bought a British one it would have a proper tail because docking the tail of this breed is illegal in all circumstances here bar medical emergency. A woman has just been jailed here for tail docking. American breeders, unfortunately,are likely to dock the tail.
Standard poodle? Different animal ! Very intelligent (up there with the border collie).Very quick to learn.(Highly likely to self-train: one of mine worked out how to open a parcel, which meant pulling, biting, tearing, tugging etc for each of the various layers and fixings of string, tape and so on.I watched, fascinated. It took her a while to discover which technique to use where but she worked it out by trial and error. The very next, identical parcel, she opened in minutes with barely a retry. She was still only a puppy). Likes a lot of mental stimulus but is not neurotic about it like some border collies. Full of life in youth and remains active and capable of bounce all its life.Not hyper though: it will lie still quite happily and doze. Grooming? A must.You can't leave the coat unbrushed for more than a day or so because the coat goes into knots and mats very quickly. It's up to you what trim you have.The coat grows continuously but never moults. Most people opt for a 'puppy [ a/k.a 'lamb'] cut' where the coat is kept trimmed to the same short level all over . I had the full show trim on mine once they were grown up . This looks magnificent. Poodles have a beautiful, proud, strutting, gait: they are real show offs. The lion mane shows to perfection on one striding out.That trim takes a little time to establish and a fair time to complete the first time. After that it requires a liitle daily maintenance until the next visit to the parlour. Poodles all bounce, as well: they have a marked pick-up in their stride, like a hackney horse.But you may find that many parlours won't do that big, show, trim in any of its variants. (You can always teach yourself how: there are books !)Again, here now it must have a tail, not some abbreviated version of a 'tail'.
Everyone knows what Labradors are like.Docile. Incapable of self-regulating at the food bowl !

Not sure what other breeds to suggest if it's principally a matter of moulting .There are plenty that don't moult much or not at all. The total non-moulters tend to be water dogs, like the poodle.