This bug is two part:
PerkSneakInvisibilityEffect [MGEF:001E3259] has Delivery set to Touch instead of Self
PerkSneakInvisibility [SPEL:001E325A] has magnitude set to 1 instead of 100
with these two changes, the sneak meter jumps to [Caution] immediately after activating sneak mode (unless standing in bright light or similar bad conditions for sneak), similar to stopping when wearing 5 pieces of chameleon armor.
In Vanilla Fallout4 the perk seems to have no effect on the stealth meter, despite several claims online, that it does. It may be, that the behavior was broken in Patch 1.6, when Bethesda fixed an invisibility bug.