Should I put a comma before and after "Luke's"?
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You're better off leaving it as-is. The commas are not needed, and would be inappropriate because they would create a false appositive, linking a plain noun (cat) with a possessive one (Luke's).
Oh, boy. There isn't really a way to do this correctly. I'd say either
(Unless they have more than one cat, in which case there's no comma.)
(The second, though correct, is rather awkward.)