It doesn't sound like you need a cover story. You've been to the midwife all the way through, so nobody could accuse you of anything if you did go to the hospital.
I would call as last minute as possible. Like have hubby call when the baby is crowning or just out unless you feel like you need to call sooner. He can sound all out of breath and tell them it came on really quickly and you're resting with the baby and waiting for them, please come quickly. Totally believable with your third child and two natural births already. Trust your instincts.
I had a private birth alleged professional, and ended up going to the hospital after she no-showed and I became concerned when I was tired. At the hospital when one person suggested I hadn't had prenatal care, I jumped down their throat. I took care of my baby every day that I was pregnant, all day, 24 hours a day. From conception to that moment. I spent significant funds on my prenatal care with nutrition and supplements and yoga and pilates and educational materials and all sorts of things. How dare they for one second suggest this baby didn't have prenatal care? There was not a word mentioned about it during delivery after my tirade.
I have a midwife team now. Between 3 people, I definitely expect one of them will show up. If I call in time. And even if I call early, they are willing to sit in a different room and leave me alone unless I ask for them. The ideal is to find professionals who are willing to respect you and your choices, but it sounds like you are in good shape to get what you want without anyone having any excuse to question you.