L-Lysine and DL-Methionine are both synthetics produced to compensate (partially) for the deficiencies of a vegetable based diet. While the DL-Methionine is likely produced via a series af laboratory reactions, chances are good the L-Lysine has been produced by gene-edited yeast, or perhaps e. coli. Calcium Carbonate obviously isn't a vegetable, you got that much right.
Methionine is an amino acid that is in high proportions in grains, but in insufficient quantities in legumes. If the chicken gets grains, methionine should be no issue. While grains are not "vegetables" per se (depending on one's definitions, but, biologically speaking grains are fruits, not vegetables), they certainly can be part of a "vegetarian" or even "vegan" (plant-based) diet. Lysine is in high quantities in Lamb's quarters, a common backyard and garden green leafy "weed." Lysine is also high in quinoa, a grain-like seed, as well as in all legumes, including alfalfa, beans, peas, etc. Lysine is in low quantities in grains, which is why people typically consider a grain plus a legume to provide a "complete protein" (having ample quantities of both of these limiting amino acids). So neither of these amino acids must be sourced from animals.
At night, the bird is kept in a small dog cage. The owner does not have any other chickens. Even if this one is too far gone, I'm still worried because this person and their friends do other "rescue" work, and what's worse they are planning on operating a chicken sanctuary. Whatever that means.
The hen's owner reveals a degree of imbalance in judgment that can nearly be predictive of certain things, e.g. how long he/she has been vegan (I might estimate at least 7 years). The greatest tragedy in this situation is not that of the hen--it is that the owner is becoming brain-damaged as a result of his/her vegan diet and consequent deficiency of vitamin B12. The longer one has been vegan, the more fanatical his or her views become. It will be nearly impossible, at this stage of the mental impairment, to persuade this individual otherwise. Vegans become irrational in certain things, most particularly in their veganism. It is often the case that prolonged vegans will consider themselves superior to others, and their own judgment more to be trusted than that of non-vegans. (Don't ask me how I know so much on this topic, but I will just say I am one who was vegan for less than a year before learning how unhealthful it was, and that I have had many close acquaintances who have followed the vegan dietary, some at the cost of their lives.)
The NIH informs us that when vegans consume large quantities of greens, their folic acid intake is in sufficiently high quantities to mask the initial symptoms of vitamin B12 deficiency. By the time they become aware of their plight, permanent brain damage may have occurred. This is no laughing matter. Without the masking effect, early symptoms of B12 deficiency usually include numbness/tingling in hands or feet, loss of short-term memory, loss of taste or smell, tiredness or exhaustion (lack of energy) for seemingly no good reason, e.g. even when one is exercising regularly and getting adequate sleep, and later symptoms (no masking for these) can progress to problems with hearing, sight, partial paralysis/difficulty walking, and finally death.
There's little hope of persuading a long-term vegan of the benefits of a different diet (and/or the harms of veganism). Vegans become entrenched in their views.
The average life expectancy of a vegan, according to a 1999 Meta-Analysis study of tens of thousands of people on five major diets (meat-eating, occasional meat-eating, pescatarian, lacto-ovo vegetarian, vegan) found that the vegans had the same mortality as did the regular meat-eaters. So while they think they are better off than the meat-eaters, they actually are not. Vegans, perhaps because they lack B12, tend to succumb far more quickly to cancer than do others. While they have much, much less chance of diabetes, colon cancer, and certain other ailments, the cancers that they do get take them down quickly; e.g. brain cancer, like a classmate of mine who died in his 30's having been vegan for at least 10 years. I have witnessed children of vegan parents who had kwashiorkor's disease. In North America. With plenty of food to eat. Again, veganism and its deleterious effects on many people is no laughing matter. There are a few, a small minority of people, who can thrive on a vegan diet. Most are not so fortunate.
If you ARE on a vegan diet, you need to know that cyanocobalamin, the most common form of B12 in supplements and as a food additive, is virtually useless. Studies have shown that 98% of this exits the body within 24 hours, whether it was taken orally or via injection (the kidneys take it out). If you are a vegan, you need to know about methylcobalamin--it is more expensive and less commonly available, but well worth it. Vegans should also be aware that nutritional/food yeast does not naturally contain B12, and it will only have it if the producer added it (probably as cyanocobalamin). There is no purely vegan source of vitamin B12--this vitamin comes only from animal products.