I am tempted to adopt a Wittgensteinian tack on this one - not saying that it is right but it seems to me an interesting approach for this type of question. For reference, my argument is based on Wittgenstein's observations in the Philosophical Investigations§193-§195.
To ask whether the proposition
(P) This coin is fair
is falsifiable is to confuse two pictures we have of coins. One is the picture we have of the coin as a symbol for all its future applications - that is to say the picture of a physical coin symbolizing an ideal random process that gives one of two results with 50-50 probability. The other picture is that of the coin as an actual physical object - e.g. made of copper, round, thin, liable to friction and tear.
Now the proposition (P) adopts the first picture - the property 'fairness' refers to the symbolic understanding of the coin as encapsulating the whole of its future application (i.e. flips) as a random process giving one of two results with a 50-50 chance.
On the other hand, the following proposition
(Q) Is P falisfiable?
refers to the second picture, i.e. the actual physical realization of the coin. And here the confusion should become apparent. (Q) asks of a symbolic property whether it is empirically falsifiable. And no such answer is forthcoming.
We are often tempted to confuse these pictures in philosophy. It seems as though 'fairness' is somehow in the coin - that it is somehow a property that we ought to be able to uncover. The reason we are "led into temptation" is because we do not take care to separate the two pictures.
The grammar of the word 'fair' is what is at issue here. What makes your question seem meaningful is that what you have in mind is the grammar of fair when we employ it symbolically - e.g. when trying to explain probability to someone. But the only grammar that actually makes the question meaningful is the practical one, i.e. the one that takes into account what compels us to call a loaded dice 'unfair' and a weighted coin 'fair' - namely measurements, production methods etc. And if this grammar makes the question meaningful, it also makes it trivial and the problem, it seems to me, disappears.