I've read this over a couple of times and am a little bit lost in understanding what you are trying to accomplish.
I think what you want to do is include the UniSelector in your web part, such that when someone clicks on 'Select' it opens a window to make the selection from a list. Once they make the selection the pop-up is closed and the UniSelector has the value. This is for the most part out of the box functionality in the UniSelector.
From what I gather your issue is that once that pop-up closes you need to do something in the web part with the newly updated value of the UniSelector. If this is the case you want to attach to the UniSelector OnSelectionChanged event in the code-behind of your web part. As the value is getting changed when a popup is getting closed you may also need to trigger a postback.