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@Colin_the_Bald
Sub-7 should be no problem, my STEEL tt fixed is only 7.6 with no weenieing.
China is your friend: you won't be sitting down much, so a carbon saddle at 70g only costs £30, low-profile carbon rims don't cost much (either buy complete wheels for ~£250 or build your own) tubulars are much lighter. £18 gets you a ~175g seatpin. I don't think I'd trust Chinese bars and stem for a hillclimb effort....
The cheapest weenie of all is GP Supersonic tubes if you're using clinchers, 100g saving over normal tubes for around £20.
3/32 drivetrain saves weight and allows you to butcher a cheap secondhand road chainset (simply file off the small ring lugs).....
Best of luck, 6 to 6.5 should be easy without breaking the bank!