The problem with keeping part of the quilt out of the water is the water is going to wick into the rest of the quilt anyway.
Your best bet is to tell the customer what happened. You HAD to press it because it was unquiltable in the condition it was in when he gave it to you. While pressing, some of the colors bled. Tell him you can either give it back to him, or you can attempt to fix the bleeder yourself. He should understand the risks 100% before giving you any kind of direction. He should also understand what the process is for fixing the bleed and give his okay for that. I also think if he has you do this work, you should charge him for it.
If he DOES give you permission to try and fix the bleeder, come back and let us know, we'll help you. I've done it several times. So far have only had one quilt that didn't rinse completely clear, and it was because I tried to fix the bleed after it was quilted.