+1 for the spring and also get a decent "floating" bridge, to go with the tremelo.
I put a Bigsby on a Stagg LS 400 (LP shape). A cheapo guitar BUT the neck is superb and with SD pups it rocks.
I don't find it de-tuning even in a live situation, but what i do find is that after 3 or 4 songs, a re-tuning is necessary, even if i don't use the tremelo. And further, the intonation doesn't suffer
But hey, isn't that part of the "appeal" of Bigsbies ???
Cheers,
Jerry