I think this is something Gauss could have proved, and the point is to come up with his sort of proof. I'm not seeing that as too hard. To show a polynomial of degree at least 1 is transcendental over K is easy. The harder step is to do that for rational functions P/Q, where the degree should be defined as the maximum of degree P and degree Q. So by rationalising we need to look at F(P, Q) where F is a general binary form (well, you should start with a monic polynomial to get to F, is what I mean). In the case of unequal degrees there is an easy reason why this can't be zero, looking at the top power of t. So we should assume equal degrees. But then P/Q can be written as a constant plus a term with unequal degrees, by long division of polynomials. That looks like the inductive step in a proof, by induction on the degree.

Presumably you would want to express the idea that K(P/Q) and K(t) are then isomorphic in some other fashion, but I don't know quite what you have in mind.