Lions can subsist on a diet of poultry, but suboptimally.

If we assume that lions' best diet is beef [1], then chicken [2] would be less optimal for them.

[1] https://tools.myfooddata.com/protein-calculator/171797/100g/...

[2] https://tools.myfooddata.com/protein-calculator/171140/wt9/1

Look at the amino acid ratios. Leucine to valine ratio is about 0.66 for chicken and 0.8 for beef. This means that protein synthesis will be bound by valine in case of chicken and what is not used in the protein synthesis will be converted to glucose and then stored as fat. Chicken will be about 80% (0.66/0.8) as nutritious as beef, judging just by two essential amino acids ratio.

Seems like it would take a lot of chickens to maintain a lion, and that would possibly require a large amount of effort for little gain compared to larger game. Lions can definitely catch chickens if there are some around and they care to.