I have some additional information. Today I noticed there was another update to the Plex client on the Shield Pro. I had just updated the app yesterday, so this is a new overnight update. Unfortunately this new version (9.05.something) did not solve the problem. This is the 3rd version that I’ve see released in the past week. I’ve been reading here on the forums today and came across this thread Nvidia Shield Dolby TrueHD playback is borked - #225 by hummelm where there’s a lot of talk about this issue which seems pretty widespread. I don’t really want to get into beta testing, but it looks like the 9.1 version may fix the issue when it’s released.
I have found a temporary solution besides Kodi though. When I checked the version of the app in the Shield by going to Settings, Apps, and then clicking on Plex, I noticed there’s an option to remove updates. I did this and it set the Plex app back to version 5.6.1.505. That version works without issue. Obviously it’s a very old version, but it works and you can even force Direct Play and Direct Stream. I really don’t understand why those are not options on newer versions. I also don’t understand how a widespread issue like this goes on for so long without being fixed, especially when there are constantly new versions of the app being released on the Google Play store (I had at least 3 updates this week alone).