Hi, I would say definitely take proof of the name change and marriage certificate etc.
In terms of logistics, if you can, park in Westfield (Shepherd’s Bush) and then walk to the White City underground station and take the central line to Bond Street. The tube ride takes about 20mins according to Google Maps
Westfield is paid parking, I think it’s about £9 if you are there for 2-14hrs and about £6 if you are there for an hour.
Good luck!