I'm hoping the warm water will assist her muscles in relaxing so she will pass that egg for you and anything in behind it. A few towels later and you'll have a clen bird who doesn't get in serious trouble! Check the appearance/scent of any material that comes out/
I'd start immediately with a shallow belly bath. Then move on, if necessary to lubing and assisting her manually. Wear latex gloves if possible. I know it's a messy procedure, but if your separate her and prepare for the mess, I think you'll be happier. I'm confident you'll do fine, the very...