You should be able to use tsmuxer to demux the subtitles into their own files (uncheck the video and audio streams first to save time and disk space, since you won’t need the individual video and audio tracks), then use mkvtoolnix to mux the subtitle files into your new encode.
That’s what I’ve tried to do with many other programs and there is no subtitle showing up in the programs, TsMuxer is probly the 6th program I’ve installed and they all only show Video and Audio, no .srt or any other format of subtitles.
It’s acting like the subtitles are hardcoded or something, yet they are removed by handbrake so doesn’t make sense to me.