I grew up in a barn like this, and I know the feeling. I grew up riding western and English, but always preferred bareback and western, (I used to be a pretty good little bareback rider
) and pretty much everyone else rode English.
While you normally shouldn't have
to talk to a barn manager about it, as it is your horse and your money, (unless of course there is some code of conduct / rule that bans western riding at your barn,) it is still her property and it would be a good idea to talk to her. Discuss with her your intentions, and the situation. While she may not be keen on it, talking to her would give you the opportunity to open her up to your ideas.
I also agree with Bsms - if she really doesn't want you to, but you really do, I think it's time to move on. If I were put in a situation where they would actually tell me to ride somewhere else, I would do so - and take my business elsewhere. Luckily for me, the worst I ever got for riding western or gaited horses were some mean words and glares, but hey, I knew I'd done nothing wrong and it was their problem.
Remember that there's always multiple sides to a situation and often several ways to handle things. I think you should choose the way that will sit best with your barn manager if you're afraid of disrupting the peace. I hope I've helped some, and good luck!