To the best of my knowledge, you can't force the site to open in the default mobile browser. The decision whether to open your site in the app's UIWebView or the default Safari browser is made by the Twitter app alone. If the Twitter app was gracious enough to provides any interface for you to choose, it might be possible, but they probably don't. Most app publishers try to keep users inside the app for as long as possible, and throwing them outside to a different app usually goes against this.
Your best bet is probably to try to improve your site's responsiveness to also work inside the Twitter app UIWebView as well (or inside other apps for that matter). If you base your responsiveness on screen width for example (CSS max-width
and friends), I assume it should also work inside the Twitter app UIWebView.
The problem you are describing is indeed a very serious and annoying one. UIWebViews inside apps don't implement by default all of Apple's special handlers (that work inside the original Safari). The app maker is expected to implement these manually, and 99% of all apps simply don't do this. It's true for many other apps as well, the Google native search app also opens URLs inside a UIWebView - and sites tend to look lousy there too.
You should also consider filing a bug with Twitter and urge them to improve website compatibility when opened inside their app.