My cumulus 200 bag, miltimat, Alpkit double bivy, tyvek groundsheet, Esbit cookset and merino thermals come to 3.5kg. 1.2kg of that is the Bivy, so 5kg is easily doable.
I think you need to order from the U.S. to get a substantially lighter bivy bag than, say, a Hunka.
Cumulus do a Pertex Endurance bivy that’s under 200g and perfect for use under a tarp, or a Pertex Shield one that’s not a lot heavier and would be fine on it’s own.
The Belgian army bivy bags come in at around 350g with the thick canvas bedroll bit trimmed off and are less than £10. They are only water resistant though so best used under a tarp.
There are lots of other options on European websites and UK postage is normally pretty cheap.