Are you sure that $G$ isn't required to be connected? I think this is needed in order to construct the "valued root datum" structure which underlies Bruhat-Tits structure theory. Anyway, the key point is that there is the concept of "valuation" on the root datum, which is really a collection of "valuations" on the possibly non-commutative groups $U_a(K)$ subject to axioms defined in the first big Bruhat-Tits paper in IHES, which I'll call BTI. The existence of this kind of structure on $G(K)$ requires the full power of the theory of the 2nd Bruhat-Tits IHES paper (developed in more "modern" terms in later work of others, such as J-K. Yu), including the connectedness of $G$, and the set of such "valuations" is an affine space for $V$. The group $N(K)$ acts naturally on this space through affine transformations, with $Z(K)$ acting through the translation formulas as you have written down. That is how $\phi$ is defined.
So what you're missing is the (highly non-trivial to develop!) definition of the principal homogeneous space for $V$ which supports the action of $N(K)$. I suppose there is a way to make the definition by bare hands (or at least give formulas, without proving things are well-defined), but my understanding (which could be incomplete) is that the affine space of "valuations" on the root datum provides the only natural way to make the definition. Take a look at section 6 of BTI to learn what a valued root datum is, and the many beautiful properties of this kind of structure. I think that BTI is more illuminating in certain conceptual respects than the Corvallis paper (though of course it doesn't have the rich supply of interesting examples as in the Corvallis paper, and is a rather challenging paper to read).