Hello,
Came here to highlight I see the same for my Panasonic TX-55JZ1500B. The app (even on a 4K TV) has a hardcoded cap of 4Mbps, two audio channels and a resolution of 1920x1080.
Fortunately (detailed below) you can work around some of the problems by logging into your Plex Media server by massaging the metadata it sends to the client.
You ned to confirm first that you can play the media without transcoding using DLNA (apt install anacrolix-dms):
dms -path /srv/media -noTranscode
It is a real shame the underlying reason is hidden away, but you can find it by configuring your TV to send debugging messages to your server (has roughly a 10sec lag before appearing in the logs) and then search in Plex Media Server.log for [MDE] Application decision which will tell you why it is rejecting the content. I saw messages telling me that it was rejecting it as the resolution was more than 1920x1080, bitrate higher than 4Mbps and there being more than two audio channels.
Fortunately you can lie to the client and massage the metadata the server sends by tinkering with the database on your media server.
Open the database containing the metadata:
sqlite3 /var/lib/plexmediaserver/Library/Application\ Support/Plex\ Media\ Server/Plug-in\ Support/Databases/com.plexapp.plugins.library.db
Now cut’n’paste:
begin transaction;
update media_items set bitrate=3900000 where bitrate>3900000;
update media_streams set bitrate=3900000 where bitrate>3900000;
update media_items set audio_channels=2 where audio_channels>2;
update media_streams set channels=2 where channels>2;
update media_items set width=1920 where width>1920;
update media_items set height=1080 where height>1080;
commit;
N.B. this probably will need redoing after every media scan and note it will show up in the user interface as these reduced values but this is only window dressing so ignore it
Once you do this, all that content just plays with no problems at full resolution and no transcoding, fortunately this is a software problem and something Plex can improve on.
I suspect Plex are somewhat tied on what they can do to the Smart TV app (and in a timely fashion) but what they could do quickly is add a feature to the media server to let the user to override what is sent to the client (ie. lie) so their media plays fine.
Now the only problem I am trying to resolve, I would love to get some help from anyone, is I am now blocked with:
INFO - [Plex for Smart TVs] [MDE] Application decision, directPlay: false, directStreamVideo: true, directStreamAudio: false
INFO - [Plex for Smart TVs] Cannot direct play with protocol: http, codec: h264. Reason: Unsupported container mkv
INFO - [Plex for Smart TVs] Cannot direct play with protocol: http, codec: ac3. Not found in profile.
INFO - [Plex for Smart TVs] Cannot direct stream with protocol: dash, codec: ac3. Not found in profile.
Really annoyingly my TV can though do all this: https://tda.panasonic-europe-service.com/docs/2z62529508z1z41680z656ez706466z25z3e03d213523275d1bb89e9d4dab549d6c1f74473/tsn3/data/ALL/TX65JZ2000B/OI/985821/TQB0E2966ENG-E.pdf (page 181)
I’m experimenting with lying about the codec’s being streamed to the TV now, but the SQL is a bit hairy to paste here.
Starting to get to the point of “well everything works with DLNA…why bother with Plex…” 
