Hi. I've been visiting airline websites looking for info on shipping a different species, but noticed a lot of warnings that airlines that take dogs would not take "snub-nosed" dogs during certain months of the year (the warmer ones). Some won't take pets if the temperature at any airport the pet would visit is above or below a certain temperature. It does seem to vary between carriers.
People do ship their dogs successfully, so, as above, you might want to check with a professional shipper to see what advice they can give you as to timing and best airline, etc.
Good luck!