No matter what, you are going to have to make a new version of the DVD. Your copies seem to be authored a little bit out of spec. You might be able to enable existing subtitles with PGCEdit, but this is not a tool I use very often, so I cannot help much. You might want to look at this guide for turning on subtitle streams by default: http://download.videohelp.com/r0lZ/pgcedit/third_party/2cool/subtitles/subtitles_on_default_pgcedit.htm
Personally, I'd use PGCDemux and extract the video, audio and subtitle streams. Open PGCDemux, navigate to the VTS_01_0.IFO file of your DVD. Make sure to put a check mark next to the video stream, as it doesn't put one there by default. Set an output area on the hard drive then hit process.
Then make a new DVD using AVStoDVD. Import the video stream which will have an .m2v extension. You import in this program by using the green + symbol on the right side of the main window. Once the video is imported, right click on it in the window and you will see options to import audio and subtitles. You'll have to do this twice to get both types of streams imported.
You can make a menu in AVStoDVD but it will be a very simple one, and won't have controls for subtitles. Your player will have to have the ability to find and play them. But the whole reason for using PGCDemux and AVStoDVD is to reauthor the movie so that the player can see the subtitles at this point.