You can use softened water when filling your tub, but there are a few concerns with doing so. Primarily, the problems are from removing all or most of the calcium from the water and increasing TDS (Total Dissolved Solids).
The danger to this is that increasing TDS may mean that you have to drain and refill more often. Simply put, your water becomes saturated with chemicals, and water balance is harder to control.
Calcium is an easier fix, because once you add more to your water, you should be fine on it for quite some time. With out enough calcium in your water, it actually becomes aggressive, searching out some other kind of 'metal' for it to feel stable. Think of it like your body, if you don't have enough calcium in your body, your body will remove it from your bones (like osteoperosis). If your tub doesn't have enough calcium, your water will begin to attack your tub, such as PVC, the shell, pump heater, etc. trying to fill this void with polymers and metals.
The best thing to do is before you fill with either water is take a sample of both to a reputable dealer who is well trained in water chemistry. I recommend BioGuard dealers, but that is because that is who we use and who I received this info from.
One other possibility is using half soft and half well water. The owner of my company does so, and his water is almost perfect by using this mixture.
Let me know if I can help in any other way. Good luck and good tubbing!