Not sure why the quality would get worse. I’ve used Handbrake to hard code/embed/burn in subtitles from SRT files to MP4 video files and didn’t notice any change in the video quality.
Is the suggested workaround the ONLY option? No. You can pester Samsung to see if they’ll fix their DLNA client to work with certain NAS devices like the My Cloud where their TV’s are having trouble streaming media. You can use a third party device like; a Raspberry pi loaded with Kodi, Amazon Fire TV loaded with Kodi, or a Roku all of which should support playing SRT files from the Twonky Media Server embedded in the My Cloud to your Samsung TV. You can load the video file + SRT file onto a USB drive and plug that into your Samsung TV. Or simply use the method above to burn in/embed the subtitles which isn’t necessarily the best workaround to your problem, but is a workaround.