
Firstly, KC registration isn't any guarantee of a good breeder, although most good breeders do register their dogs with the KC, so do a lot of puppy farmers and everyone inbetween, so it's up to you as a puppy buyer to do your research and try to find out whether a breeder is the sort you would like to support. And it's entirely up to you what your interpretation of a good breeder is, although there are a few obvious things that would tell you someone isn't such a good breeder, there are also a lot of grey areas as well. Such as health tests, I can think off the top of my head some very old and reputeable kennels who are doing very few health tests, because they just haven't historically, but since we've developed more and more relevant health tests, you would hope they would be used to try and make more informed breeding decisions. I can also think of ABS members who have done things that aren't in the spirit at all of the assured breeder scheme, but they have no problem selling their pups, and being a member of the KC's AB Scheme probably does help them to do this. So it's a big mine field I'm afraid.
Secondly, health checks means nothing, some breeders use this term to confirm their vet has said their dog has four legs, a waggy tail and a wet nose; there are a number of health tests for Labradors, my preference would be for both parents to have hip scores around or below the breed median of 9, and nice and even, so 5/5 is good, 9/0 isn't as good. Both parents should have elbow grades of 0, and both should have a current, clear BVA eye cert (within the last 12 months). And at least one parent should be tested clear for PRA, CNM, EIC, HNPK and SD2; if one parent is clear, then none of the pups can be affected and go on to possibly develop the condition. However, if one parent is tested as carrier or affected, then the other should be tested clear. Having a dog that is a carrier or affected status for a condition doesn't (to me) exclude them from breeding on from, as long as the dog is healthy (affected status does not mean they automatically suffer from the condition, just that they could potentially develop it), but this should be explained to potential puppy buyers.
You may have to widen your search, I've had a pup go to Germany (the owner came over to choose her puppy then came over to collect her) and have pups in Kent and Surrey in the UK, and have people wanting pups from me from across the UK although I have no breeding plans within the next two years (I've only had three litters in total in eight years of breeding). A lot of breeders have been inundated with enquiries at the minute, and not all reply to every enquiry whether or not they have pups, or have any planned litters. Rather than look at breeders close by to you, look through the breeders section and single out those whose dogs you like, who are doing the health tests, and if you like what you're seeing about them and their dogs, and their breeding plans, then contact them explaining who you are, why you're looking to add a puppy, what you like about their dogs etc. Another way to look for a potential litter is to look through the stud dog section, again, look at their dogs, what you like about them, have they got all the health tests and good results, and contact them asking if they have had any visiting bitches, and would they mind putting you in touch.
Good luck with your search.