Concerning a test, the ugly result would be reproduced when calling fig.set_size_inches((7,4)) after creating the secondary axes and before drawing/saving it.

Added test - fails on master, passes w/ this PR

Good catch.
I think the standard-ish way to avoid the fact that inverted() is "eager" would be

    bb = mtransforms.TransformedBbox(
        mtransforms.Bbox.from_bounds(*rect),
        mtransforms.BboxTransformFrom(parent.bbox)
        + mtransforms.BboxTransformTo(parent.figure.bbox))

    def secondary_locator(ax, renderer):
        return bb

    return secondary_locator

which also works, and avoids constructing new transform nodes at each call (#12031).
Or alternatively, use Transform.transform_bbox which constructs "dumb" bboxes that don't keep track of their transform stack.

That makes there be no secondary axis rendered at all (or I guess its located off page?)

I'm pretty sure parent_bbox has to be the draw-time bbox, not the locator instantiation bbox.

Indeed, sorry.
I think such an approach could be made to work, but in the meantime this looks like a good fix.