Carefully grab the hard lump of poop, and slowly pull it off, starting at one side.
Yes, your fingers will get poopy.
Yes, the chick will hate it.
Yes, it will pull the fuzz off the chick's back end.
If you are careful, it will not actually rip the skin of the chick.
(No, I'm not suggesting this to be cruel. Even going slowly and carefully, it's faster than soaking. And by pulling off the fuzz with the poop, there is less for new poop to stick to. I would not do it if the poop is stuck all over the back half of the chick, but usually the stuck poop is only attached to a small area of fuzz.)
If that sounds too awful, go back to soaking or using a wet cloth. Think about how warm a baby chick is, and try to use water that warm so the chick doesn't get chilled. Maybe 100 degrees fahrenheit, if you want to check with a thermometer.
Chicks always hate having their butts cleaned, but there can be a point when it needs doing or else the chick dies. It sounds like you might be at that point. Of course, you could just cull (kill) the chick, but that seems a bit drastic at this point.