Does renormalization make quantum fields into (slightly) nonlinear functionals of test functions?

Field theories are nonlinear because the quantum fields satisfy nonlinear dynamical equations.

But renormalization does not make quantum fields into a nonlinear functional of test functions. The Wightman distributions are, by definition, linear functionals of the test functions, and Wightman distributions always encode renormalized fields.)

Instead it changes the space of test functions to one where the interacting quantum fields are perturbatively well-defined. This gives a family of representations of the field algebra depending on an energy scale. All these representations are equivalent, due to the renormalization group, and the corresponding Wightman functions are independent of the renormalization energy. (In simpler, exactly solvable toy examples that need infinite renormalization, this can actually be checked.)

The dependence on the energy scale would not be present if contributions to all ordered were summed up (though nobody has the slightest idea how to do this nonperturbative step). The energy scale is simply a redundant parameter the influences the approximations calculated by perturbation theory.

The renormalization group is an exact but unobservable symmetry (just like gauge symmetry) that removes this extra freedom, but as computations in a fixed gauge may spoil gauge-independence numerically, so computations at a fixed energy scale spoil renormalization group invariance numerically.

Note that Wightman functions are in principle observable. Indeed, the Kadanoff-Baym equations, the equations modeling high energy heavy ion collision experiments. are dynamical equations for the 2-particle Wightman functions and their ordered analoga.

[added 22.01.2018] In the above, the renormalization group refers to the group defined by Stúckelberg an Bogoliubov, not to that by Kadanoff and Wilson, which is only a semigroup. See here.

This paper by Borcherds offers a dense treatment of perturbative renormalization in an altered version of Wightman's axioms. There is no failure of linearity in the test functions anywhere - see especially the bottom of page 8.

I believe the problem in your reasoning is your main assumption that test functions define absolute energy scales. At least in the context of Borcherds's treatment, Feynman measures are essentially made from the data of energy cutoffs (which form an infinite dimensional space of possibilities), and the Wightman distributions are constructed from a choice of a Lagrangian and a Feynman measure, so the energy scale is an input to the distribution. One may then probe the distributions with test functions, but these test functions are not connected to the energy scale that you chose when constructing the distribution. In fact, there is an infinite dimensional ultraviolet group (also called the group of renormalizations) that acts both on the space of Lagrangians and on the space of Feynman measures, such that the resulting Wightman distributions are fixed. This group does not act on the space of test functions.

G. Scharf, instead of renormalizations, splits arguments in the distributions. I think it is equivalent to replacing one of distributions with a regular function.