Some options to consider:
(1) Can your satellite box decode the captions? If so, you might be able to go back to using component cables. If not, you might be able to change the satellite box to one that can do the decoding.
(2) What about using s-video? The would improve quality a bit compared to composite video but should still allow the TV to decode captions. I have not done a side-by-side comparison between component video and s-video to know how much difference there is.
(3) When you switch back and forth between composite and component video can YOU see a difference in picture quality? Depending on the source material and screen size the difference may or may not be obvious to everybody. If YOU are satisfied with the picture quality using composite video then it does not really matter whether component is technically superior.
(4) Is it too late to consider returning the new TV and choosing a replacement that does decode captions from a component signal? If you notice a significant worsening of the image using composite or s-video and if your satellite box does not decode captions then this may be your best option.
Unfortunately, all TV purchasing decisions involve trade-offs of price, quality, features ... so there may not be a perfect solution.