You specify that
the vertical and the horizontal antenna cables are connected together at the middle point (in parallel, center-braid braid-center and to the feeder cable)
Swapping the two conductors of a transmission line (in the cases where it makes sense at all) produces a 180° phase shift, not a 90° one. Therefore, this antenna design will not produce circular polarization.
If your antenna design is based on identical perpendicular elements, one of those elements must have a 90° phase shift applied such as by a delay line (1/4 wave extra length of coax).
If you did add a 90° phase shift to one element it seems like it could work but I'm not sure what the effect of having the two ends of differing voltage meet at a corner would be. I think at that point you would be better off using antenna modeling software to get a simulated answer.