My friend had a similar experience during Beautiful with two groups of people getting into a fight during the show. One was talking loudly and talking pictures, the others two rows behind got annoyed and told them to stop, which made the talking group get annoyed back. My poor friend was caught in the middle and totally had the the 1st act ruined. She asked to be reseated at intermission, after explaining the problem to the usher, and they promtly did.
On the other end of the spectrum, my "favorite" story. I was in London seeing Top Hat for the 2nd time and had a seat in the 2nd row center. I loved the show so during curtain call I was giving a standing ovation, not being totally aware that this is not as much a thing in the UK as it is here, and not many people besides me stood up. The elderly woman behind me grabbed the back of my pants and pulled me down back into to my seat. Seriously. That happened. I was so perplexed I couldn't even react.
Her husband... and several other people sitting nearby... apologized to me on her behalf after the show, but man, way to ruin the exhilaration of just having seen a wonderful show!
To the OP, I think it would be better to call Roundabout (I'm not sure you would even get a hold of the house staff at the AA) and explain the problem, perhaps focusing on them training their staff better to handle such situations. Because there really isn't much they can do now to help you with your experience. Even if they were to offer you tickets for another performance (unlikely), you probably wouldn't be able to take up that offer, and even a refund (again, unlikely) wouldn't really help because those folks still ruined the show for you. So you should think about what you're trying to achieve before you get in touch with them, and it might be easier to speak to them over the phone, instead of just writing a letter/email.