For completeness, I'm going to provide functions for generating the circumcircle and incircle of a spherical triangle:
sphericalCircumcircle[pts_] := Module[{cen, cosr, nrm},
cen = Total[Cross @@@ Partition[pts, 2, 1, 1]];
nrm = Norm[cen];
cosr = Det[pts]/nrm; cen /= nrm;
BSplineCurve[Composition[TranslationTransform[cen cosr],
RotationTransform[{{0, 0, 1}, cen}]][
Sqrt[(1 - cosr) (1 + cosr)]
{{1, 0, 0}, {1, 1, 0}, {-1, 1, 0}, {-1, 0, 0},
{-1, -1, 0}, {1, -1, 0}, {1, 0, 0}}],
SplineDegree -> 2,
SplineKnots -> {0, 0, 0, 1/4, 1/2, 1/2, 3/4, 1, 1, 1},
SplineWeights -> {1, 1/2, 1/2, 1, 1/2, 1/2, 1}]]
sphericalIncircle[pts_] := Module[{cen, sinr, nrm},
cen = RotateLeft[Map[Norm, Cross @@@ Partition[pts, 2, 1, 1]]] . pts;
nrm = Norm[cen];
sinr = Det[pts]/nrm; cen /= nrm;
BSplineCurve[Composition[TranslationTransform[cen Sqrt[(1 - sinr) (1 + sinr)]],
RotationTransform[{{0, 0, 1}, cen}]][sinr
{{1, 0, 0}, {1, 1, 0}, {-1, 1, 0}, {-1, 0, 0},
{-1, -1, 0}, {1, -1, 0}, {1, 0, 0}}],
SplineDegree -> 2,
SplineKnots -> {0, 0, 0, 1/4, 1/2, 1/2, 3/4, 1, 1, 1},
SplineWeights -> {1, 1/2, 1/2, 1, 1/2, 1/2, 1}]]
Expressions for the appropriate radii and centers were derived through standard spherical trigonometry. Both of these functions use the NURBS representation of a circle (see e.g. Piegl and Tiller), and both assume that the spherical triangle is determined by three unit vectors, corresponding to a spherical triangle embedded in a unit sphere centered at the origin. (You should rescale and translate appropriately for spherical triangles on spheres with different centers or radii.)
As a concrete demonstration, generate a random spherical triangle:
BlockRandom[SeedRandom[1337, Method -> "MersenneTwister"]; (* for reproducibility *)
tri = RandomPoint[Sphere[], 3];]
Visualize the triangle and its circumcircle and incircle:
(* http://mathematica.stackexchange.com/a/10994 *)
arc[center_?VectorQ, {start_?VectorQ, end_?VectorQ}] := Module[{ang, co, r},
ang = VectorAngle[start - center, end - center];
co = Cos[ang/2]; r = EuclideanDistance[center, start];
BSplineCurve[{start, center + r/co Normalize[(start + end)/2 - center], end},
SplineDegree -> 2, SplineKnots -> {0, 0, 0, 1, 1, 1},
SplineWeights -> {1, co, 1}]]
Legended[Graphics3D[{{Opacity[0.6], Sphere[]},
{Directive[Red, AbsolutePointSize[6]], Point[tri]},
{Directive[Blue, AbsoluteThickness[3]],
arc[{0, 0, 0}, #] & /@ Partition[tri, 2, 1, 1]},
{AbsoluteThickness[2],
{ColorData[97, 3], sphericalCircumcircle[tri]},
{ColorData[97, 4], sphericalIncircle[tri]}}},
Boxed -> False, ViewPoint -> {-2.4, 1.3, -2.}],
LineLegend[{Directive[AbsoluteThickness[2], ColorData[97, 3]],
Directive[AbsoluteThickness[2], ColorData[97, 4]]},
{"spherical circumcircle", "spherical incircle"}]]

vertices
is a list of three 2D coordinates representing your vertices, thenInsphere[vertices]
gives you the 2D sphere (aka circle) representing the inscribed circle. You can extract the center (the first part) or the radius (the second part). I haven't reverse-engineered your code, and I don't understand "build an inscribed circle into a triangle on a sphere", so I don't know how you want to extend this to 3D. $\endgroup$