When my mother made borscht, she added Sour Salt, which is just crystals of citric acid. I don't know if it's still available in the spice section, because my jar (French's brand) is still fine after maybe 20+ years.
I like the idea of using pickle brine, especially if it's dill pickles; gets the flavor without changing the color. To me, the color is the best part, and adding anything that could muddy it would make it much less appealing (sorry, chrose! borscht is borscht and schav is schav and never the twain shall meet, at least not in my kitchen
)