I put a B7 on my Tokai ES 120 (335 clone), I'd say it added my 3/4 of a pound to the weight. As far as string tension...it does increase the string length quite a bit, but I haven't noticed any difference in the tension...if there is it's a very small amount.
The Bigsby is an aluminum casting, so it doesn't add much weight. I put a B5 on my SG, and I don't notice any difference. The Bigsby requires somewhat longer string length, but has a tension bar where the stop tail piece sits, so the tension is the same.
I mounted this with a Vibramate plate. One is available to fit the B7 to the Les Paul. It saves drilling/filling holes in the guitar.