Complex search algorithm - Points on a sphere - Best approach
Let me start by suggesting a brute-force method, then suggest some ways to speed it up:

Pick a pair of (high-weight) points $P_0,P_1$ in the sector.  For each pair of points $Q_0,Q_1$ on the sphere, guess that maybe $P_0=Q_0$ and $P_1=Q_1$: align the sector (rotate &amp; translate it) so that $P_0$ is on top of $Q_0$ and $P_1$ is on top of $Q_1$.  There is a unique way to do this.  Then check your candidate placement of the sector, by mapping the sector onto the globe and looking at all the other points in the sector to see how well they match the points on the sphere.  If you enumerate all pairs of points $Q_0,Q_1$ on the sphere, you're guaranteed to find the correct placement of the sector for at least one of them.  So, this algorithm works, at the cost of being slow: it takes $O(n^2)$ time, where $n$ is the number of points on the sphere.

You can dramatically speed this up in several ways.

First, you can filter by the distance $d(P_0,P_1)$ between the two points.  Instead of enumerating all points $Q_0,Q_1$, only enumerate pairs $Q_0,Q_1$ such that $d(Q_0,Q_1) \approx d(P_0,P_1)$.  How do you do that?  You enumerate all possibilities for $Q_0$, and for each such $Q_0$, enumerate all points $Q_1$ that are at distance approximately $d(P_0,P_1)$ from $Q_0$.  The latter can be done using a data structure for nearest neighbor search.  If you choose two points $P_0,P_1$ that are close to each other, probably there will be only a small number of points $Q_1$ at that distance from $Q_0$.  So, heuristically, the running time might become much closer to $O(n)$.

Second, you might be able to filter using the weights.  You could pick a pair of high-weight points $P_0,P_1$ in the sector, and only consider points $Q_0,Q_1$ whose weights are above some threshold.  You'd have to experiment how to choose that threshold.  Or, you could enumerate the points $Q_0$ in sorted order of the weight, highest weight first.

You can increase robustness to small errors in the location of the points in the sector by repeating the above multiple times for multiple different choices of $P_0,P_1$.