Again, not sure about the LG, however the Shield absolutely should work on some of them.
If you look at my post in the big thread, where I give my test results, I highlight a number of settings that are set on all my kit… The TV, the Shield and my AVR…
I’d suggest that there are going to be a number of settings you are going to need to check on both the Shield, and your LG TV… On the LG, check the settings for the HDMI ports and see if there is the ability to set them to something like “Enhanced”.
Also do a general google search on enabling Dolby Vision for your LG TV.