Short answer
Yes, cats can safely eat a small amount of plain, cooked lobster meat. The cooked flesh is a fine occasional treat. However, raw lobster carries a risk of bacteria or parasites that could make your cat sick, and the hard shell is a choking hazard that can also cause intestinal blockages if swallowed. Never give your cat lobster that's been cooked in butter, garlic, or other seasonings, as those additives can be harmful. Stick to a tiny piece of unseasoned, fully cooked meat as a rare snack.
Lobster — Food safety facts
Safe to share · Cats · full breakdown on the linked food page
Plain cooked lobster meat is safe. Never give the shell or drawn butter.
The full verdict, portion guidance and serving notes are collected on the Lobster food facts page.