With the benefit of hindsight of two labours (second one over 10lb, natural delivery) I would push for another growth scan, and if they are not happy to give you one then a discussion with the consultant - I assume you are consultant led - to come up with a scenario plan. This will help you relax and know what you are doing, which will help you going into the birth, however it happens.
My agreed plan went something like this: allowed to go to 40+4 for natural delivery, after which I had a provisional c-section booked to prevent the baby becoming too large (which all agreed would happen if I went any more over due). I categorically did not want to be induced, which they were surprisingly fine about.
I was measuring big all the way through with DC2, lots of people told me about the inaccuracy of scans, but she was, indeed, huge. I delivered her with barely a scratch so it certainly can be done, but she was my second which I think makes a significant difference. My first labour with a much smaller baby was protracted and ended up with emergency forceps. Nothing to do with the size, just a lack of progress.
I think you are wise to ask more questions, if only to set your mind at rest that all the thinking, as you say, is 'joined up'.
Good luck!