Eliot's characterization of these as "[n]ames that never belong to more than one cat" indicates that they are nonce words: words that are coined "on the spot" to be used for a single purpose, and presumably will never be used again. By their nature, nonce words tend to be nonsensical coinages that are not derived from existing words—after all, if the word has a clear derivation it's possible that someone might recoin it independently in the future using the same logic, which is usually the opposite of what the writer would want.
Nonce words often "sound real" because they are often constructed using familiar English affixes or tense forms (for example, Edward Lear's "runcible spoon"), or are designed to evoke familiar words. "Munkustrap" brings to mind monkeys, minks, muskrats, and other things, which helps explain why it sounds "legitimate" and sends you searching for a derivation. But it's probably safe to assume it's just nonsense, and thoroughly runcible nonsense at that.