How do you find a good crossbreed breeder?
Essentially the same as you would a decent breeder of pedigrees; by doing a lot of research and being prepared to wait and/or travel for a puppy from the right breeder. Obviously you don't have the benefit of a breed club to start from but other than that it's very similar.
This list of questions to ask a breeder was written with pedigrees in mind however it absolutely all still applies regardless of whether someone is breeding crosses, mongrels or an as yet unrecognised breed.
A decent breeder will want to satisfy themselves that their bitch is a good breeding prospect beyond just thinking she's a lovely dog. They should have a very clear idea of what they want to produce, have evaluated their bitch's conformation and temperament as objectively as possible and then picked a stud dog who they feel is the absolute best match for her in order to achieve their breeding goals. Ideally they should be seeking some kind of independent evaluation of their bitch, whether that's via competing in dog sports/activities, working (if applicable for the breed), showing, PaT qualifications or even something as simple as the KC Good Citizen Dog Scheme awards.
They should have a good understanding of conformation and how it affects the dog. They should be confident their bitch has excellent structure, not in a "for the breed" type way but in terms of being structurally sound, and have picked a stud dog who they hope will improve upon her weaker points. They should be aware of potentially health-impacting conformational trends within the breed and be actively breeding away from these, which is still important with crosses as breeds of a similar type can share the same or similar conformational issues.
They should take health testing very seriously, this is particularly important if you're looking at CKCS crosses. The breed can be affected by a raft of different health issues and these can't all be avoided by simply crossing them with another breed. This list covers some of the major issues but by no means all, if you want some more in depth reading have a look at this site.
The most major issues in the breed which need to be carefully considered even when cross breeding are Syringomyelia (SM) and Mitral Valve Disease (MVD). There's no simple DNA test for either of these conditions and they can't be completely avoided just by crossing a CKCS with another breed, particularly as both conditions have been seen in other breeds which are commonly used to cross them with.
If a breeder is producing CKCS crosses they should still be utilising testing schemes and breeding protocols for SM and MVD for the CKCS parent.
For SM this involves an MRI scan to check for any sign of the malformation of the skull which causes the condition and ideally the dog should be graded a 0. Personally, if they're graded any higher, given the debilitating nature of SM, I'd want to see evidence of a 0 grading for the other parent if they're of a breed which can be affected by either SM itself or lower grades of the malformation.
Current MVD breeding protocols recommend not breeding any CKCS until they have reached the age of 2.5 years with no detectable murmur AND both that dog's parents are each free of murmurs at 5 years.
If you're looking specifically at the CKCS x Bichon then both parents should have current BVA eye tests (which are repeated annually) as both breeds can be affected by hereditary cataracts and no DNA test is available for either. Both can also be prone to luxating patella so I'd want to see evidence that both dogs had been properly screened for that. Finally, both should have had a DNA test for Macrothrombocytopenia with a clear result.
There are also several DNA tests available for the CKCS which, although for conditions not present in other breeds (and so CKCS cross puppies couldn't be affected) I'd expect a conscientious breeder to have done them anyway so they know if there's a chance they could be producing puppies who are carriers. These tests are for Dry Eye/Curly Coat syndrome, Episodic Falling syndrome and Muscular Dystrophy.
With regards to the DNA tests it is acceptable for a breeder not to do them providing they have documentation showing that both parents of the dog in question were tested and had clear results. This is known as being "clear by parentage".
There are health issues in all breeds which are potentially heritable but it either hasn't been confirmed yet or the nature of how they're inherited hasn't been discovered. Many breeds share some of the same conditions and so any breeder, whatever they're producing, should have a thorough knowledge of the lines they're using and whether there have been any signs of potentially heritable health issues in them.
If the bitch is KC registered it's a good idea to get her full name and do a check online to find out how many (if any) registered litters she's had. It's not uncommon for less than ideal breeders of pedigrees to start breeding crosses once they've registered the maximum number of litters allowed per bitch. You can do this via the KC's MyKC service, you do need to sign up but it's free and incredibly useful for checking these things.
A breeder being licensed by their local council is generally a bad sign, the vast majority of councils only require this when breeding at volumes that no decent breeder ever would. Good breeders generally don't breed very often, they want to see a litter grow then consider how they can improve things for their next litter. Having multiple litters on the ground at once, especially if they're different breeds/crosses, or breeding several litters a year are definitely red flags.
Obviously the puppies should be being raised in the home as part of the family and they should be clean, have no eye/nose discharge and have well trimmed nails. There should be very clear evidence the litter is inside full time; look for things like a whelping box or pen set up, potentially a litter area (depending on the age of the puppies), toys and objects for the puppies to play with and explore, etc. The breeder should also be taking a very active role in raising the pups, it's so much more than just letting mum get on with it until she's sick of them then entertaining them with a few toys. There's loads a breeder can (and should) be doing to prepare their litter for life in their new homes. This site gives you an idea of the minimum a breeder should be doing during each developmental stage. Puppy Culture is the absolute gold standard of puppy raising, you can get the basics on that site and there's also an active FB group where you can see the sorts of lengths PC breeders go to in order to maximise the chances of their puppies being happy, confident and well rounded individuals.
The absolute best advice I can give is to look for a breeder, not a puppy. A good breeder, who is putting their all into breeding healthy, sound puppies of excellent temperament, who will help and support you at any time during the life of a dog they've bred is absolutely worth their weight in gold. A puppy will hopefully be part of your family for well over a decade, it's so worth putting in the extra effort now and being prepared to wait for the right litter to stack all possible odds in favour of your puppy living a very long, happy and healthy life with you.