EDIT: I misread the question and thought you wanted clarification on what it meant. Ah well, I already wrote it, so I'll leave it up in case anyone else finds a use for it.
I don't know what he meant by saying 'a great winter warmer' (maybe it's a British word for 'sweater'?) but saying there was 'a bit of a bite' to the winter meant it was a good temperature throughout the season, and even when it got noticeably colder at times, it was still considered great or it wasn't cold enough to make the winter seem anything less than great (depending on what the speaker meant; he might have liked the bite or was just mentioning that there were a few exceptions to enjoying that winter).
It's like saying something spicy 'has a bit of a kick'; you can taste the spice but it isn't considered unpleasant. Instead, the spice adds to the flavour - or it's meant as a warning to anyone who doesn't know there's spice in the food.
Oui, merci à tous les deux!
C'est vrai que "bite" peut se référer au froid "mordant", je n'y avais pas pensé. Mais on dit aussi que le froid pique les joues, donc "avoir du piquant" me semble une bonne idée.