As the original poster of this topic, I am happy to report that I was able to stream a full game this past Sunday without issues. Whether this is an isolated experience or not remains to be seen because I haven't attempted since (or before for that matter), but I do know that the app built-in to the TV still has not been updated and is still listed as being the version from May 2015. This is obviously the same version that I had problems with last year, but just to be safe, I have my AppleTV on standby. If the issues resume, I will let you know.
Also, I'll agree with DrMagnusCranium that neither party will admit fault. When this started to happen last year I contacted LG first and they told me it was an MLB.TV issue, then I contacted MLB.TV and they told me it was a LG issues so at that point, I just resorted to other media players, etc to stream the games.