What would you do?

I am currently 39+3.

I'm 4cm dilated, 2cm long and cervix is very posterior. Lost a lot of mucus over the last few days and strong false contractions after a sweep last week. Baby is very high still and not engaged at all.

Been having painful Braxton Hicks for weeks.

Baby is very large, measuring 42 weeks last week and has a very large head.

I'm trying for a vbac. I'm concerned that his size will lead to another c-section as my first child had a very large head and was stuck. She also never engaged even during labour before the decision for a C-section was made.

I'm having another sweep at the end of the week and will be induced at 41 weeks if no luck by then.

I do not want to do over 40 weeks due to his size. I want to give myself the best chance at avoiding another C-section.

Unsure if inducing at 40 weeks will increase or decrease my chances.

Would you ask to be induced before 41 weeks? Or stick with the current plan?

Edit: Yes doctor is happy to induce at 41 weeks for vbac