Wilson's perspective is intrinsically approximate; changing the scale changes the theory. This is independent of renormalized perturbation theory (whether causal or not), where the theory tells us that there is a vector space of parameters from which to choose one point that defines the theory. The vector space is finite-dimensional iff the theory is renormalizable. In the latter case we pick a parameter vector by calculating its consequences for observable behavior and matching as many key quantities as the vector space of parameters has dimensions. This is deemed sufficient and needs no mathematical axiom of choice of any sort. Instead it involves experiments that restrict the parameter space to sufficiently narrow regions. In the infinite-dimensional case the situation is similar, except that we would need to match infinitely many key quantities, which we do not (and will never) have. This just means that we are not able to tell which precise choice Nature is using. But we don't know this anyway for any physical theory - even in QED, the best theory we ever had, we know Nature's choice only to 12 digits of accuracy or so. Nevertheless, QED is very predictive. Infinite dimensions do not harm predictability elsewhere in physics. In fluid dynamics, the solutions of interest belong to an infinite-dimensional space. But we are always satisfied with finite-dimensional approximations - the industry pays a lot for finite element simulations because its results are very useful in spite of their approximate nature. Thus there is nothing bad in not knowing the infinite-dimensional details as long as we have good enough finite-dimensional approximations.