Looking at src[-1] would work but feels like poor form. My advice is stop trying to make do ... while work here, it isn't looking good.

I feel like more idiomatic iterating through a string tends to loop on src not being a terminator.

    while (*src)
    {
       *dst = *src;
       ++src;
       ++dst;
    }

    *dst = '\0';

I feel like this is idiomatic C but needlessly verbose. Most people would combine the increment with the assignment. And most people would recognize putting it in the while condition as a common strcpy.