Only if you can find a Shield Pro, with the 500GB internal drive, which was recently discontinued by Nvidia.
The Shield is more than capable of transcoding 1080p video. However, the 16GB Shield does not have enough internal storage space for the PMS database, metadata, etc. Right now it is not possible to move the data to external storage. Plex and Nvidia are working on a solution, but there is no announced delivery date.
Check out systems with a 7th or 8th gen Intel processor, i3/5/7-7xxx & i3/5/7-8xxx (Kaby Lake & Coffee Lake). You can then take advantage of the iGPU and use hardware accelerated transcoding. The iGPU in the 7th & 8th gen systems can handle H265 in addition to H264. If you really want to dig into the details, see Datasheet, Volume 1, Processor Graphics section for each generation at Technical Resources: Intel® Core™ Processors.
Nvidia GPUs can also be used for hardware accelerated transcoding. Be aware that Nvidia limits their consumer GPUs to two simultaneous transcodes. I’ve no direct experience with AMD GPUs & iGPUs, so cannot speak to their transcoding capabilities.
On a more general note…
As long as media is Direct Played, Plex does not require much CPU power to stream a movie. The Celeron in my NAS can easily stream Blu-ray rips at 20 - 40 Mbps (i.e. rip Blu-ray to mkv file w/ MakeMKV).
Before buying a new PMS system, investigate why your media is transcoding (container not supported, audio/video codec not supported, media encoded at incompatible profile & level, etc.). Many clients show the transcode reason onscreen. You can also look at Status -> Now Playing via the PMS web interface.
Look for ways to avoid transcoding in PMS: remux from mkv to mp4; process media with Handbrake or similar; possibly change clients to one that supports more formats; etc.
It is not always possible to avoid transcoding. However, if you are transcoding everything, throwing more CPU cycles at the problem is not necessarily the best solution. Better to make transcoding the exception, not the rule.