Nope. You can chose either 24fps or 23.976fps.
And if the issue was related to the HDMI frame rate, the drops would be consistent in time instead of happenning randomly.
I've experienced this in multiple Shield models and devices. I suspect the issue is rather in how Android implements video decoding or maybe how OS stuff in the background might interfere in video decoding. Something weird is that I've never seen this in Plex, only in Netflix and other streaming apps.
I tried using the refresh frame rate app, or the beta match frame rate feature, but these never worked consistently. Sometimes the fps was great but the audio was out of sync by like 2 seconds, sometimes the video kinda froze and ran at 1fps, etc.
Ah, my apologies. I misunderstood you all this time, then.
Strange issue indeed.