They should be neutered or better yet CULLED!
Why is neutering not sufficient? Should not being able to hunt be a death sentence?
A basset that cannot perform what they were ORIGINALLY breed for should not be in the show ring or even worse breed
Many people have found that once given the opportunity and some experience their show hounds CAN do what they were originally bred for. BUT most of us do not have to opportunity to try it. The basset trials seem to be in very limited areas. There are certainly none near me. I have nowhere that I can let my dogs run free to chase bunnies.
However, when a couple of my guys did spot a bunny on the lawn, they were after it like a shot, opened up beautifully and nearly caught it. They come from many generations of show-bred hounds.
On the subject of Labradoodles, while they originally were supposedly bred with a purpose (a non-shedding guide dog with the temperament of a labrador), the purpose failed. Most labradoodles DO shed and their appearance and temperament varies widely. The VAST majority of people breeding labradoodles and other "designer breeds" are doing it for the big bucks these puppies bring, not actually doing breed development.
Honestly, with something like 700 (I think) breeds in the world, 153 of which are registered byt the AKC, I'd be hard pressed to believe that you can't find one to do the job you have in mind, but if that's the case then a careful, planned breeding program to produce a breed that fits the bill would be a valid endeavor IMO.
And as for "bright companion dogs with the solid disposition of good labs", we already have those, they're called Labrador Retrievers.