I want to discuss this because I afraid that the answer is no. In SR we stuck with the transformations from Poincare Group because this transformations leave invariant the exact form of the Lorentz Metric tensor. Any other transformation will change the components of the Lorentz Metric Tensor and screw up all the calculations that we used to do. Because of that we can not introduce an accelerated or rotated coordinate system by some strange transformation of coordinates. For the same reason we can not do even Galilran Transformation (see attached file). Suppose we have a coordinate system with arbitrary metric tensor: gmn where the indexes indicate coordinates and can take values: m,n = 0,1,2,3. If we take 2 points: P1(0,0,0,0) and P2(x0,x1,x2,x3) and connect them with a straight line then the invariant length of this line will be s2=gmnxmxn where we have 2 contractions. In t-x plane it will be: s2=g00(x0)2+2g01x0x1+g11(x1)2 (this shows how contractions work). For a resting clock (x1=0) we have: s=x0√g00. The reading of a clock will reflect the invariant length s and will be not equal to the reading of time coordinate x0. The square root is a “time factor” (see attached article). The same way there exist “x-factor”, “y-factor”, and “z-factor”. The simple measurement is possible only if the factors on all axes equal to 1. That is: g00=|g11|=|g22|=|g33|=1. This is true for Lorentz Metrics. Einstein’s “Frames” can represent the corresponding coordinate systems only if all the axes factors equal to 1.