Feasible to build your own WebKit-based UIWebView for iOS?

This is very feasible. In fact, it has already been done. Impact is an iOS game engine that compiles its own JavaScript core. Here's a blog post with some technical details.

I won't elaborate further here, the links should provide more than enough detail.

It is possible, but not allowed by Apple. For the same official reasons that they did not allow Google to ship their own WebKit version with their browser. Google has to use UIWebView.