I do agree on the big bivvy being stronger in heavy winds, unless you peg an umbrella down with its back to the wind you could be in for a starry night...
Overwraps for an oval aren't that big or bulky to carry. Mine is around 45cm long with maybe 10-15cm across and fits in the rod sling pocket with my umbrella.
They aren't new, but trying looking at JRC Contact bivvies and brolly systems. Even though they are claimed as brollies, the framework looks more like outer poles than a real brolly