A native reader may have more definitive understanding but my suggestion is to imagine
(1) the statement with and w/o the adverb and
(2) how the speaker would have said it and what words he might have emphasised: eg "I let you catch me" vs "I let you catch me easily", or perhaps "I neatly (conveniently?) let you catch me".
I think まんまと can be used with 罠にかかる by itself so it adds to the irony of the speaker's statement, reminding us how we was in control all along. Without the adverb it becomes very factual, with the adverb we get a better insight into the speaker's intention, and given the situation, he is likely to use one to express his feeling.
Just for reference I also looked at the following sample sentences from the "プログレッシブ英和・和英辞典 to confirm my understanding of まんまと
I was neatly taken in.
I won't let you get away so easily.
In that way he succeeded in enticing the monkey to approach.