Mark, who gives several examples, uses it only for human hair, not animal hair or fur. But Gaelic colours also refer to shade as much as hue. I would use liath for a very pale grey rabbit, or pale blue if there were such a thing, and glas for mid-grey to bright blue. But it is not a precise science.
Also, liath has only recently become a general-purpose colour, so that would be another reason for not using it for something that would have been known for a long time.
There's only 12 basic colours in Gaelic, + a few used only in specific contexts. That means you sometimes have to be a bit flexible to describe a particular rabbit. So my solution would be to say liath as you suggest but you will never get everyone to agree. To me liath gives the sense of 'very light' without giving the sense of 'blond' that bàn would give. In addition any perceived human-hair metaphor would be fine as my uncle had hair that colour before it went snow white.
My point about how long the animal has been around is important too. If this were a species of rabbit that had been around for centuries, like the mountain hare, it would have a name based on what colours used to mean, like the madadh ruadh 'red fox'. But since that is not the case, it is quite reasonable to describe it using the modern palette.