At how "Fast" does it become unstable? And since when has instability stopped us from doing anything? We play around with things that are unstable, just to see what would happen. That's why we have half-life numbers for all sorts of materials, because they are unstable. That doesn't mean we can't make it a controlled instability. Just because a Warp Field would become unstable and uncontrollable at Warp six, doesn't mean it won't be unstable BUT controllable at Warp five.
Also, Solar Radiation is something we have to overcome just to leave the Terra-Luna system. Once we figure that out, this Hawking Radiation should be a snap.