That one is easy. First let's clarify what exactly we mean with bigger. Let's say we mean larger in volume. Now we pick a procedure for determining the volume of animals, say we submerge them briefly in a large water tank and mark the resulting water level. That animal that lead to the highest water level is the bigger one. Any objections?
The same objections you have regarding the fair coin.
You say you submerge the animals in a tank of water. How does that work? You have this giant tank, put in the animal and one molecule of water? Just like a single toin coss, a single molecule of water will not tell you much.
You need hundreds of billions of water molecules?
Ok. And how do you know these will behave in the expected way?
Because you have a model in your head how molecules behave? Well, I have a model how fair coins behave.
Because you saw different sized objects result in different water levels before? I saw fair and unfair coins result in different head:tail distributions before.
But similar to your objection about the Bayesian view of probability requiring money and betting to be defined, you've now defined the idea of size to be dependent on giant tanks of water
What does it mean that an elephant is bigger then a dog?
If you think you know the answer, you are probably wrong.