If you're tuning to C#... I would suggest leaving your guitar as a hardtail. You're just asking for set-up problems if you're talking about tuning that low... AND adding a tremolo. A wammy pedal would work nice.

However, to satisfy your curiousity. As was said earlier. Les Pauls and SG's come with Bigsby's all the time... Some models it's a standard feature. See the Black Beauty for example... That's probably the only LP I would ever buy... Aside from a Slash signature model. Bigsby makes kits that can be installed on arch top or flat top models... check www.stew-mac.com You should be able to find them pretty easy on there.

I don't have much experience with tremolos... specifically the Khalers. I have some experience with FR's and bigsby's but I don't use a lot of trem in my playing... And the only electrics I own are strats fitted with the Vintage style trem... very basic. Your heavy string gauges don't fit well with tremolos... but any decent luthier or tech should be able to make it work if that's what you want. If it's something you want to pursue... check stew-mac out and go consult a tech/luthier... they should be able to answer your questions. Take your guitar in so they can specifically look at it and gie you a straight up answer.