I had 2 BPP/NST's done in my 42nd week (i got to 41+4 and had them at 41 and 41+2) and both times they found a perfectly happy baby, a perfectly functioning placenta and nothing to worry about. I had my homebirth. She was not at all over-cooked, i think i just cook a little longer than the traditional 40 weeks.
I can well understand, after your last birth, that panic might be setting in, but really i cannot see it panning out like you worry it will.
I think you will go for the BPP and find all is fine. IF, it is a HUGE IF, they find a reason to induce, unless they are the quietest hospital in the world or your baby is in real and immediate danger they will at least send you home to pack a bag and ask you to come in later that night/the next day. Which means when you get the fake phonecall about the fake family crisis that you have to go attend to you will be at home and can then call the hospital and tell them "sorry, this cannot wait, i'll call you tomorrow and we'll reschedule" and thus keep yourself out of there for another few days. Unless of course the baby really IS in trouble (and you can have them detail what is wrong and what it means and get a 2nd and 3rd opinion and STILL disagree if tey're being intervention-happy about it) in which case this is moot.
I was really worried about "monitoring" a they called it, and felt my homebirth slipping away, but honest and true, it actually put everyone caring for me at greater ease, and made my homebirth MORE attainable.