Could it also be a
Baby? (A single English word.)
The Roman gods hid treasure
The Roman God Jupiter was only a newborn when he was hidden by his mother Cybele.${}^1$ (She wanted to protect him from being swallowed by his father, Cronos.)
${}^1$Although this is Greek Mythology, Jupiter was the Roman name of Zeus and Cybele was the Roman name of Rhea.
In robes of rose and green
Jupiter was hidden in a cave on Mount Dicte, part of the Lassithi Mountains on the east of the island of Crete, Greece. Parts of the island are green, also where the Red Cretan Tulip grows.
A stone lay at the center
A stone lay${}^2$ at the center of a christening.
Every Christian baby must be baptised; they must undergo a christening. At the center of the word "christening" is sten which can be derived from the word stone (by removing the letter that lays at the center, funnily enough).
${}^2$Note that by using the word lay instead of lays, the phrase is in past tense, since a christening only happens once.
And gold was in between
The baby would be baptised by a priest who would most likely be wearing robes of gold.
Fetched by a dog named rover
A reference to the song Hush, Little Baby via the lyric, "Mama's going to buy you a dog named Rover."
Lee in "Show Me" was seen
There is an actress that goes by the name of Pui Fan Lee. She is part of a children's television series called, "Show Me, Show Me". (In the children's television series, "Teletubbies", there is a character called Po who is also voiced by her.)
Sir, move if you are clever
When babies become cleverer, they begin to crawl and move from one location to another.
And tell me what I mean
Could you mean a baby?
Title:
Hints Among Confusion
The riddle would be confusing if baby was the answer, and the acronym HAC also stands for Hospital-Acquired Condition, which might relate to giving birth to a newborn in hospital since pregnancy is a medical condition.