Are you talking about the HDD (disk/storage) or the CPU/processor of your Plex Media Server?
Given your description I consider this is about Plex trying to transcode the 4K content to meet some restriction of the TV — e.g. if the file container, video/audio/subtitle codec are not supported or the bitrate exceeds some threshold… In your case, Plex will do that transcode using only the CPU and can be quite demanding… too much for some systems.
To dig into this…
- what CPU are you using in this machine (exact model)
- can you post some of the technical specifications of the video that fails to play? → see Investigate Media Information and Formats | Plex Support
- can you confirm your client is set to original/maximum video quality / direct play is enabled
If the issue is actually a limitation of your disks… do you have an option to free up some space / add more storage? 
PS: as for ruling out the network… keep in mind most current videos use variable bitrates, with the max bitrate easily exceeding the average by 40-60% (there’s examples of max bitrate peaks at double the average rate)… also… with 4K material this can result in massive data volumes to be transferred.
Hence: would you mind sharing how you connected server and TV to your router (e.g. wired vs. wireless, bandwidth…)
Edit: useful background on playing 4K material… [INFO] Plex, 4k, transcoding, and you - aka the rules of 4k
