I am by no means an expert on the subject, but you might be able to, at least in theory, preserve sharp corners in a monochromatic pseudo-SDF if you used either a Bilateral filter a Directional Bicubic filter instead of a standard Bilinear filter. Unfortunately, I don't have any means of testing my theory currently, and just have been unable to find any documents showing similar tests to what I am describing.

Alternatively, if you don't mind using a second channel, then you could also try having one channel for Horizontal Distance and another for Vertical Distance, and use a Difference of Laplacean (DoL) energy pyramid to compress the texture so that redundant information wouldn't be recorded.

A third and final theoretical solution would be to experiment with a Hexagonally Sampled Texture via Array Set Addressing.

I will link all of the relevant articles where I got my information/ideas shortly.