Sounds great! What you are looking to do, we have done with keyboards like CS-Pinyin and Yi on Windows, and Chinese, Japanese and Korean on the web. However, the solution right now is a bit of work, because we don’t have as smooth a pathway for picker-based input methods as with our normal model of input.
You can have a look at https://github.com/keymanapp/keyboards/pull/64 for an example of a picker-based input method; this one runs only on Windows. We have not yet published the source for our picker-based keyboards for web but these will be available in the future.