I agree that breeding the gold-and-black hen to the gold-and-white rooster will produce at least some chicks that are gold-and-white. That should be either half of chicks or all of chicks. Depending on which genes the rooster carries, half of the chicks may be gold-and-black.
If that hen and that rooster produce just gold-and-white chicks, breed a son back to his mother to get some chicks that show black with their gold.
If they are suffering from inbreeding, that will probably get worse if you breed a son back to his mother, so I can't say whether it will be a good idea in this particular situation. Maybe try it and see what happens? But as regards the color genetics, it is the easiest way to get more gold-and-black birds.
For starters, I would keep trying. If you set more eggs, you might find that some chicks do fine in each color, and you would find those by hatching larger numbers of eggs. You could use two incubators, and set every egg that is laid: once a week the eggs go in the incubator, once a week the ready-for-lockdown eggs move into the other incubator to hatch. You'll just have time to clean up after one hatch before the next week's eggs are ready for lockdown.I have been trying to hatch some and so has a friend but so far only black ones seem to survive more than 3 days. I currently have 6 more from my mini flock in my bator so. The ones we have lost all stayed off ok but quickly grew weak, stopped walking and couldn't even sit up. By day 3 they no longer opened their eyes and they died soon after. I tried feeding them honey water and vitamin b mixed with warm yoghurt with a syringe and this seemed to help a little for a short time.
Since what you fed them that seemed to help a little, I wonder if the parents have a deficiency, which leads to deficiences in the eggs and the chicks? Offering a vitamin supplement to the parents might help. If it does help, it will take a while to see results: parents eat supplement, hens take days or weeks before they form better eggs, eggs incubate for three weeks, chicks hatch (hopefully without problems.)
If all the other chickens are doing well, but this particular breed is showing problems, they may have different needs than the other chickens. Maybe they really do need more of something. You could ask the breeder what she feeds, and see if that is different than what you have been feeding. If hers do better than yours, she must have it right. Or if hers do badly as well, then either it's a problem with both sets of feed, or else it's not a food problem after all.
What you see could be a symptom of inbreeding. You could try hatching half-appenzeller chicks by crossing with other hens or roosters, and compare the results. If inbreeding is the problem, mixed chicks should do better than the purebreds, which might help with identifying the problem.
It may not completely rule out inbreeding. If someone imported just a few birds into the country years ago, and all the Appenzellers are descended from those few, the entire breed may be badly inbred in your country, even without breeding close relatives in the past generation or two.
But if the entire breed IS badly inbred, you don't have many choices. You could live with poor hatch/survival rates and breed from the healthiest of what you get, or introduce another breed and then start the long process of breeding back to the standard, or try to import more birds from another country (probably not practical), or give up on the breed. None of those is quick and easy.
If only the blacks are doing well, you could mix the blacks with the other colors, then try to breed back to those other colors. You will get lots of poorly-colored birds that way before you get any good ones, but you should at least be able to keep the other traits (body size & shape, temperament, laying abilities, etc.) No matter what is actually causing the problem, if the blacks do not have it, that could be a good starting point. I do not think there is anything in the actual color genetics to cause trouble, based on what colors exist without trouble in other breeds, so it is probably something else that happens to be in that line of birds that is not directly responsible for the color. You might be able to breed away from whatever that is (specific genes or inbreeding in general) and end up with a healthy strain of the colors you want.
