I'm afraid I haven't the time to do this properly, (life, grr) but it appears you can write a modified Brun's sum like this
C_2(X)/log x + m*integral(3,X) C_2(t)dt/(t*log^2 t)
where C_2(X) is sum(p twin prime>2 log p/p) (ie log(3)3+log(5)/5+log(11)/11+log(17)/17+log(29)/29 +...)
and m=1 (this then=1.058... like we discussed ten years ago)
now what's interesting is you can choose m to be less than one, and the limit then appears to be O(log^-2 X), as X->inf. for certain m,say m=1/(4*C_twin) and C_twin=.66016...
sorry I can't explain better, life gets in the way all the time.