I wonder if "bringing" is gerund or participle.
I don't want you bringing your dog with you.
Gerund. It's the complement of "want". What don't you want?
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Gerund. It's the complement of "want".
What don't you want? You bringing your dog with you.
Or you could explain it as a catenative structure: want ... -ing— want [someone] (doing something). As such you might say it's an alternate version of want ... to .... (I don't want you to bring your dog with you).
CJ