If you've already looked into the indexes in your manuals and none of them mention closed captioning, and I assume you have. Then you may need to look into hiring a professional service. So far as closed captioning being something simple to do, I beg to differ. Closed captions are encoded into the vertical blanking bar of analogue video. And even though you are doing digital editing, the signal that goes to a viewer is an analogue signal. Since my roots are in linear video production, I know how tube cameras generate the vertical blanking interval. And for the closed captioning we put on county government meetings, it was done with an analogue device downstream of the switcher, just before entering the modulator for delivery to the headend.
So closed captioning on an NLE has to be placed by a program that has access & control over the vertical blanking interval, which I don't think digital video even has. Until it is converted to a composite signal, there would be no need to create a vertical blanking interval inside the digital file. But since I've seen players with CC options, so there must be a way for it to be stored in the digital file. But I'm sure you'd need specialized software to make unique vertical blanking intervals.
And there's the expense. It is very specialized software, with a very small market. Just to break even it would have to be quite expensive. So you are most likely to make it a reasonable expense by shopping around for someone to do it for you. I'd check with your local professional video equipment suppliers to find who can do closed captioning in your area.
Good luck. And thank the government for requiring closed captions on music programming.