I noticed something in the Room of Harmonic Convergence in the Lost Logbooks run that COULD be the half pendant.

I don't think this is a half-pendant. The thing you've indicated is too small in my opinion to be the half-pendant.
But I think I may have found it: (click on the photo to make it bigger)

I think that thing in the box just above the clock is it.
Alec, could you possibly look at this episode during the temple run with 1:11-1:09 left and see if you can get a better picture of the object I boxed?
Not in perfect quality, but here's the Lost Lariat half pendant:

This is definitely the half-pendant. Chris even picks it up as he walks by the Throne.
TSM, please update the OP to reflect that this half-pendant was "On the Throne in the Throne Room"