Yeah. Unfortunately, I ended with what I realized was my weakest point. I deleted the line, but then I realized you had already responded. Sorry about that.So you can't use Lorentz transformations. So what? Any physical quantity you could ever calculate will give you the exact same result in either coordinate system.
An example of a physical quantity, by the way, would be the time required for a light beam to bounce off some far-away mirror and return.
However, since you brought it up, though you make the case that you "can't use Lorentz Transformations" that is a cop-out. Changing the metric does not release you from the Lorentz Transformation--it only changes the form of the Lorentz Transformation.