That's the plan. Just waiting on the Tune It to arrive and then the Bigsby will get a thorough inspection, including removal and disassembly. If there is any friction or resistance at all, it will get attended to.If lubes don't work, I'd take the strings off and check if the Bigsby "free falls" When you lift it and drop it in place. It should face no resistance when it falls. Maybe it's a bit out of alignment or may need a shim. Just a thought.
I use Tri-Flow for the slots in my guitar`s nut. This stuff is incredible. A super-lubricant.
But I would not use it on my Bigsby because I would fear that it would catch-up dust or dirt.