250ppm is a near-perfect amount of calcium hardness to have in a spa. I would stick with unsoftened water unless it is very inconvenient to bypass the softener. Are you sure about the "utilizes no salt at all" part? From the page you linked - "It simply regenerates with 2 boxes of common table salt in less than 30 minutes, and provides the owner with soft water up to 1600 gallons (or up to ~40 days)." Generally the ion exchange resins adsorb calcium and magnesium ions, then when recharged with salt the sodium ions are preferentially adsorbed onto the resin, displacing the Ca/Mg that are washed away in a backwash cycle, then the sodium is exchanged for Ca / Mg during use, and then the cycle repeats.
I think you'll be fine with either softened or unsoftened water, just plan on re-adding enough calcium to get back up above 200ppm if you fill with softened water.