The BGH1 is actually a bit closer to having full V-Log, since it has a modified version of V-Log L with 8-stops in the shadows and 5-stops in the highlights (13 stops total), as compared to say V-Log on the EVA1 which has 8-stops in the shadows and 6-stops in the highlights (14 stops total).
The thing is that the Micro 4/3 sensor in the camera is really only capable of capturing 13 stops of dynamic range anyway, so I'm not sure the full V-Log would help that much. The 12-stop version of V-Log L on the GH5S did seem to be artificially constraining that camera somewhat, but I don't know that the same is true for the 13-stop version in the BGH1. For example, the Blackmagic Pocket Cinema Camera 4K which shoots RAW using the same sensor, only claims 13 stops of dynamic range as well.
There is a comparison between the Panasonic GH5S, BGH1, and S5 in this video:
