I have a Gibson Les Paul (1980) where I built a Bigsby on.
To be honest, I have hated it, it did not keep on tone.
I put graphite bridge sadles on it, replaced the nut with tusq, let the bigsy and guitar several times check at by good techs. But still this very big problem with keeping tune.
And I was almost wanting to get the the bigsby off, when I read that Joe B used Nut Sauce for his Les Pauls with Bigsby.
And I must say, brilliant!!! It solves all issues I had. Guitar stays in tune, even with heavy usage of the bigsby!
I think that when you have tuning issues with a guitar, you really should give this a try.