Intriguing problem :)
Here's a sketch of an idea that I've never tried...
If you let the origin of your flatland
map to the north pole of your sphere,
then you could view the length of any point
P from the pole as an arc length on the sphere,
and a subtended angle could be defined as
|P| |P|
a = (-------- 2 pi) mod (2 pi) = ----- mod 2 pi
2 pi r r
where a is the angle subtended, r is the radius
of the sphere, and |P| is the distance of P from
the origin in flatland.
If we rotate around a vector orthogonal to the
direction of P in flatland, this would give us a
mapping dependent only on the position of P.
If (0,0,1) is your north pole vector, for
example, the rotation axis would be
axis = (0,0,1) x P/|P| = (-P.y,P.x,0)
The idea is to carry out your rotations and
translations normally in the plane; then, when
you want to move to the sphere, apply the angle
axis transformation. In other words, just "bend"
the current orientation onto the sphere in the
direction of P.
Hmmm, I've probably missed something...