<p>I have a JSF form that needs to upload a file via an icon click (i.e. without showing the textfield/browse button of the file upload component). </p>

<p>Is it possible to trigger the file-open dialog and uploading functionality of the upload component via some other component type? </p>

In my opinion, it's never too late to adopt a better practice - or to drop a worse one - so I'd say "Yes, you should start".
However ... (there's always a "but") ...
... one of the biggest gains of TDD is that it impacts on your design, encouraging you to keep reponsibilties separate, interactions clean and so on.
At this point in your project, you may find it difficult to get tests written for some aspects of your framework. Don't give up though, even if you can't test some areas, your quality will be the better for the areas you can test, and your skills will improve for the experience.