Pets

...no matter what breeders or the AKA tout.

Physiological characteristics like a particular texture and/or color of coat, body morphology (size, conformation), eye color, etc. are easily verifiable and become a 'breed standard' because they involve very plastic expressions in the genome - they are comparatively 'simple'.

Behavioral predispositions are to some extent breed-consistent in terms of things like herding, guarding, types of prey-seek (gazehound vs scenthound, etc.) but they have been selected for much, much longer than breed registry organizations and standards have been around. If an Anatolian tribe member who bred dogs for their extended family's herd a thousand years ago noticed the offspring of a particular dog and/or bitch were really good at herding, that dog and/or bitch would definitely be bred again, regardless of whether the offpring's tails had the proper curl. If a breeder of Anatolian Shepherd Dogs today found pups in a litter without the proper tail curl, they'd be neutered and sold as pets (by a 'responsible' breeder.)

"Temperament" is the least consistent and reliable of all claimed breed characteristics, and the most influenced by early conditioning and experience of the individual dog.

If what you like about Bichons or Wheatens is the size or the coat type, ask some of the larger shelters to notify you when dogs surrendered or brought in have similar types of coat/size.

But if what interests you is the individual personality of the dog, you may find it in a dog that looks nothing like either a Wheaten or a Bichon. You might find it in a rare, one-of-a-kind dog that is absolutely unique in lineage, morphology and behavior, and totally committed to being YOUR best and most loyal and loving friend.

Whichever you choose, best of luck in making the connection, and thank you for providing a forever home for a representative of one of the best species on earth!

thoughtfully,
Bright