There is an assembly at the base of your luber/sizer that adjusts how deep the bullet enters the die.
If this is not backed out all the way, so the entire bullet fully enters the die, sizing nose first won't reduce the nose diameter, especially with your .366 die.

This LAMII has the liftout mechanism and the depth is set with the nuts on the long rods on the sides and a threaded plug at the bottom of the press. [Linked Image]
I lube only in this press with slightly oversized dies or with bullets that barely size.

This LAMI has the mechanism removed (cut off) to operate like a Star sizer, but with conventional dies. If a bullet does not travel its entire length through the very bottom of the die, it will not be sized at the point that the die is the smallest diameter (at the very bottom) because they are generally cut with a taper.
[Linked Image]

I'd also ask Veral why he's now recommending a bore-ride nose...

Since these dies are sizing about .001 larger after spring back, a .365 die might get you the .3665-.3670 range which will be larger than groove, yet still apparently closer to the diameter your throating reamer is making.

I just don't think you are getting to the maximum travel sizing in your .366 die.