OP, correct me if I'm wrong:
1. When you told FIL you would be having the baby at a birthing centre, it was the truth.
2. You have since changed your mind and are planning a home birth.
3. You do not need FIL to babysit your older child because s/he is going to be mostly present and there will be a doula there to help look after him/her.
4. FIL does know your due date but there is no guarantee you will actually deliver anywhere near that date.
My advice? Don't bother saying anything at all. Don't call him when labour starts; there is no reason to do so, since he isn't needed for child care. Have your baby at home with your DH, midwives and older child. Once everyone is cleaned up and ready for visitors, call your FIL and invite him over to meet his new grandchild. At that point, you can tell him that you knew he would be uncomfortable with your change in birthing plan and didn't want to stress him or yourselves out and as he can see, everything worked out just fine.
This is basically what a friend of mine did, minus the older child and minus parents within walking distance to visit. She decided on a home birth and knew her parents would freak so she didn't tell them until after the baby arrived. Of course, both sets of parents were a 5 hour flight away so it's a bit different.