At that point it would no longer be closed caption. It would be subtitles.
That can be done with the built-in titler, or if you want to make it easier on yourself, you might choose to do it in After Effects.
Here is something you might try, using your completed cc file..
I wanted to use open captions, which are helpful for certain audiences, but Pr does not have that function. So I opened my cc file in the source monitor, and made the SM as large as possible. Then I did a screen cap of the cc file. Importiing that screen cap into Pr I put it in the timeline, sized it for my project, and keyed out the background. I exported the timeline and the cc is always there no matter what I play it on. Worked very well for me, and not many keystrokes or much typing to do, just waiting for the scc to finish playing.
There are also scc --> text apps, but I haven't tried any.
Is there any way to embed closed captions inside the video so that closed caption show up when movie is played irrespective to specific player ?
Thank you, Jerry
What program are you using for the screen capture??