No, I doubt he would still be alive. He was too close to Harry, which made him a threat to Voldemort. He would have died somehow.

Sirius might not have had the mirror with him, in which case, Harry would have judged him to be in danger.

I think that when you are desperate to do something, you don't stop and think about all the things you could do. Harry was panicing when he saw that Sirius was in trouble, and he used the method he first thought of. So there was a very small possibility that anyone would have thought of that. Maybe even JKR forgot about it? Who knows...