Sorry to hear both of you are put in this situation...it sucks and I don't think either of you have done anything beyond Boeing SOP's.
I offer this only from my own experience, take it or leave it. I hope it helps someone who reads it...
VNAV path is good only when little change is expected in the wind on descent and/or when your cruise mach is well below Mmo. (say 0.76)
If you have a really strong headwind or tailwind I would always use VNAV speed. VNAV path, although good, never gives you that protection quickly enough. If I have 100 kts tailwind component at upper cruise levels, I descend early by 10% of that TWC. (ie 10nm) in VNAV speed. If its a HWC, 10% late. (I just use time to measure it - ALT HOLD before TOPD then when the chinese glideslope appears, start the timer...If you are doing say 480kts GS or 8nm/min, then start you VNAV speed descent 1min 15 sec after the slope appears, for the headwind, before for the tailwind.
This my own rule of thumb and everyone has a different way to do it of course but I find it picks up the profile by 15-10,000'. Then use VNAV path from there as most of the height restrictions on a star are generally below 10,000'.
It never is an exact science and I'm constantly learning new things. Hope it all works out...