You need to go by car with an animal across the tunnel; you can't take it on the ferry as a foot passenger and you can't take it on the Eurostar (probably except an assistance dog). An acquaintance of mine faced this problem when trying to repatriate his lost cat back from Paris.

You need to meet the rules precisely: rabies vaccine a suitable time ahead, veterinary check in France, etc. However, given those, it's straightforward to take a dog, cat or ferret [1] through the channel tunnel; they have a dedicated pet checkin area. If you need a vet recommendation in the area, I can ask my mother, who does this trip with her dogs a couple of times a year. You will, in practice, need to stay overnight in Northern France or Belgium to do the veterinary formalities.

Word of advice: check very carefully that the animal's microchip number is recorded properly on the paperwork. It will be handwritten or transcribed several times and you will have a lot of trouble if the number on the various records do not match.

[1] All of whom have a rabies vaccine available