I would suggest that the size of the dog matters considerable, not for rentals as much as for the airlines. If the dog is a service dog and large you are fine but beyond that it would be best if it was small enough for a carrier under the seat. If it is a larger dog it will go into cargo. If that happens it cannot be too hot or too cold in flight, at which times the airlines will not then carry the dog. That plus all of the International Vet paperwork that you pick up before you get on a plane.
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