I use the kitchen sprayer. I hold the chick much like you have it in the picture and spray until the poop loosens and comes off without picking it at all. If there is a big dangling clump you can crush it between your fingers as it softens but don't pull it, crush and rinse. I test the water temp on my arm much like one would do for a baby's bottle. I try to keep the rest of the bird dry and have a heat lamp ready for it to go under (basically back into the brooder).
Check your brooder temp. Pasty butt is caused by a brooder that is running on the warm and dry side and they are drinking more to compensate. A brooder should have enough room to have a gradient of temperature from 70-85 degrees. The chicks like to go warm up under the heat and go play in the cooler zones.