Having any word there is kinda redundant. You've already mentioned being "caught" or "captured" at the beginning of the sentence. There's no need to say it again at the end.
I'd go with "like a rat in a trap". I've heard that used as an idiom before. Even if it's not the most popular, well-known idiom, it's fairly easy to guess what it means (it is fairly self-explanatory).
To me, "like a rat caught in a trap" would imply struggling. The "caught" in there gives the impression that the rat is stuck, struggling, trapped.
Having the "being" implies present tense, or more the process of being caught rather than the end result - capture - which is what you're focusing on. If you were to use "like a rat being caught in a trap" anywhere, that context would be in describing the process of a capture.
Before he could move, the police descended on him. In less than a minute, he had been caught like a rat in a trap.
Like a rat caught in a trap, the man struggled ferociously as he was overwhelmed by the officers.
Like a rat being caught in a trap, the fleeing man found himself surrounded. "You're under arrest!" blared a megaphone.
All of these have different connotations and are used in different contexts. So what your specific case is is going to change the answer here.