Why doesn't accelerating reference frames in QFT lead to horrible paradoxes? Background
So I remember that in Special Relativity while one can define acceleration things can go horribly wrong has happened historically (I'm sure there many other paradoxes). The real reason of things going wrong is while in special relativity on can talk about acceleration in a limited sense it does not make sense globally.
Question
How is it that in QFT (Special Relativity + Quantum Mechanics) we do not arrive at any sort of paradox like in the case of special relativity talking about QFT due to this? 
P.S: I am aware this question is similar to: Equivalence Principle holding in Special Relativity? (let alone QFT) (However, I am of the opinion to know if it were a duplicate I'd need an answer for the previous one)
 A: Let's go into the details of the definition of QFT in particle physics, more than quantum mechanics and special relativity combined.

QFT treats particles as excited states (also called quanta) of their underlying fields, which are—in a sense—more fundamental than the basic particles. Interactions between particles are described by interaction terms in the Lagrangian involving their corresponding fields. Each interaction can be visually represented by Feynman diagrams, which are formal computational tools, in the process of relativistic perturbation theory. 

It is a mathematical tool for particle interactions , and acceleration has no meaning within this framework, because all interactions happen through virtual particles according to the appropriate interactions, strong , weak or electromagnetic. You cannot accelerate an electron in  QFT, you can write a Feynman diagram for its increase in energy with an interaction of a photon.
The basic fields are represented by plane wave solutions, Dirac for fermions, Klein-Gordon for bosons, quantized Maxwell for photons. Creation and annihilation operators operating on these fields give the interactions and Feynman diagrams the tools to calculate them.
Everything is Lorentz invariant.
Please note that quantum field theory is a calculational tool that has been used for other frames than particle physics, where wavefunctions can be defined.
So  accelerated observer frameworks  with respect to the interactions happening do not affect the calculations using QFT in particle physics, i.e. crossections, lifetimes etc.  The QFT  calculations are done in the center of  mass system of the particles, and results can be translated to other frames without having to involve QFT. 
Accelerated observer frames will not affect the calculations for measureable interactions between elementary particles. 
The effect of observer acceleration on the QFT vacuum is being studied in new theories , for example here,

In this article,  we have presented the theory of accelerated quantum electrodynamics and used it to explore the radiation produced by uniform accelerated motion

but no contradictions with data measurements are found, or paradoxes.
