Taking up Feyre's comment, here is my solutions so far:
I split the SliceContourPlot3D into three individual plots, one per cross section. The individual pieces are:
pdr = Import[ "https://cdn.spacetelescope.org/archives/images/newsfeature/heic0601a.jpg"]
texfunc1[x_] := 1/(1.1 - x^.5)
texfunc2[x_] := 0.7 Sin[25 x] + .5/(1.1 - (x)^.5)
texfunc3[x_] := 1/(x + .1)
Row[{
SliceDensityPlot3D[texfunc1[Sqrt[x^2 + y^2 + z^2]], {"CenterCutSphere", \[Pi],3/2 \[Pi]}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, Boxed -> False, Axes -> False, Lighting -> "Neutral", ColorFunction -> "Rainbow"],
SliceDensityPlot3D[texfunc2[Sqrt[x^2 + y^2 + z^2]], {"CenterCutSphere", (3 \[Pi])/2, \[Pi]/4}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1},Lighting -> "Neutral", Boxed -> False, Axes -> False,ColorFunction -> "Rainbow"],
SliceDensityPlot3D[texfunc3[Sqrt[x^2 + y^2 + z^2]], {"ZStackedPlanes", {0}}, {x, -1, 1}, {z, -1, 1}, {y, -1, 1},RegionFunction -> Function[{x, y, z}, Sqrt[x^2 + y^2 + z^2] <= 1], Lighting -> "Neutral", Boxed -> False, Axes -> False],
SphericalPlot3D[1.02, {u, Pi/2, Pi}, {v, 0, 2 Pi}, MaxRecursion -> 0, TextureCoordinateFunction -> ({3/4 #5, 1/2 + #4/2} &), PlotStyle -> Directive[Texture[pdr], Specularity[White, 50]], Lighting -> "Neutral", RegionFunction -> Function[{x, y, z, u, v, r}, 0 <= v <= 3/4*2 \[Pi]], Mesh -> False, PlotPoints -> 50, Boxed -> False, Axes -> False],
SphericalPlot3D[1.0, {u, 0, Pi/2}, {v, 0, 2 \[Pi]}, MaxRecursion -> 0, TextureCoordinateFunction -> ({#5, #4/2} &), PlotStyle -> Directive[Texture[ImageTake[pdr, {1, 250}, {200, 400}]], Specularity[White, 50]], Mesh -> False, PlotPoints -> 50, Lighting -> "Neutral", Boxed -> False, Axes -> False]
}]

Putting everything together into a single function:
clump3SlicesPlot3D[{{texfunc1_, label1_, colscheme1_}, {texfunc2_,
label2_, colscheme2_}, {texfunc3_, label3_, colscheme3_}}, texture_, opts : OptionsPattern[{SliceDensityPlot3D, SphericalPlot3D}]] :=
Module[{tmaxr, maxVal, minVal},
maxVal =First[NMaximize[{#@x, 0 <= x <= 1}, x]] & /@ {texfunc1, texfunc2, texfunc3};
minVal = First[NMinimize[{#@x, 0 <= x <= 1}, x]] & /@ {texfunc1, texfunc2, texfunc3};
Show[{
SliceDensityPlot3D[texfunc1[Sqrt[x^2 + y^2 + z^2]],
{"CenterCutSphere", \[Pi], 3/2 \[Pi]},
{x, -1, 1}, {y, -1, 1}, {z, -1, 1},
ColorFunction -> colscheme1,
Evaluate[FilterRules[{opts}, Options[SliceDensityPlot3D]]],
Lighting -> "Neutral"],
SliceDensityPlot3D[texfunc2[Sqrt[x^2 + y^2 + z^2]],
{"CenterCutSphere", (3 \[Pi])/2, \[Pi]/4},
{x, -1, 1}, {y, -1, 1}, {z, -1, 1},
ColorFunction -> colscheme2,
Evaluate[FilterRules[{opts}, Options[SliceDensityPlot3D]]],
Lighting -> "Neutral"],
SliceDensityPlot3D[texfunc3[Sqrt[x^2 + y^2 + z^2]],
{"ZStackedPlanes", {0}},
{x, -1, 1}, {z, -1, 1}, {y, -1, 1},
ColorFunction -> colscheme3,
RegionFunction -> Function[{x, y, z}, Sqrt[x^2 + y^2 + z^2] <= 1],
Evaluate[FilterRules[{opts}, Options[SliceDensityPlot3D]]],
PlotPoints -> 50,
Lighting -> "Neutral"],
SphericalPlot3D[1.02, {u, Pi/2, Pi}, {v, 0, 2 Pi},
MaxRecursion -> 0,
TextureCoordinateFunction -> ({3/4 #5, 1/2 + #4/2} &),
PlotStyle -> Directive[Texture[texture], Specularity[White, 50]],
Lighting -> "Neutral",
RegionFunction -> Function[{x, y, z, u, v, r}, 0 <= v <= 3/4*2 \[Pi]],
Evaluate[FilterRules[{opts}, Options[SphericalPlot3D]]],
Mesh -> False,
PlotPoints -> 50],
SphericalPlot3D[1.02, {u, 0, Pi/2}, {v, 0, 2 \[Pi]},
MaxRecursion -> 0,
TextureCoordinateFunction -> ({#5, #4/2} &),
PlotStyle -> Directive[Texture[texture], Specularity[White, 50]],
Evaluate[FilterRules[{opts}, Options[SphericalPlot3D]]],
Mesh -> False,
PlotPoints -> 50,
Lighting -> "Neutral",
Method -> {"ShrinkWrap" -> True},
PlotLegends -> {
Placed[BarLegend[{colscheme2, {minVal[[2]], maxVal[[1]]}},
LegendLabel -> label2, LegendMarkerSize -> 200], Below],
Placed[BarLegend[{colscheme1, {minVal[[1]], maxVal[[1]]}},
LegendLabel -> label1, LegendMarkerSize -> 200], Below],
Placed[BarLegend[{colscheme3, {minVal[[3]], maxVal[[1]]}},
LegendLabel -> label3, LegendMarkerSize -> 200], Below]}],
Graphics3D[{
Text[Framed[Style[label1, 15, Black, Bold], Background -> White],
CoordinateTransform["Spherical" -> "Cartesian",
{1.1, (3 \[Pi])/4, 0.1 \[Pi]}]],
Text[Framed[Style[label2, 15, Black, Bold], Background -> White],
CoordinateTransform["Spherical" -> "Cartesian",
{1.1, (3 \[Pi])/4, -1/2 \[Pi] 1.1}]],
Text[Framed[Style[label3, 15, Black, Bold], Background -> White],
CoordinateTransform["Spherical" -> "Cartesian",
{1.1, 0.9 \[Pi]/2, -1/4 \[Pi]}]]}]
},
SphericalRegion -> False,
Boxed -> False,
Axes -> False,
ViewPoint -> {1.8592398366973455`, -1.666975163782372`, -2.283510681597605`},
ViewAngle -> 0.5011114127587017`,
ViewVertical -> {-0.6000864995229751`, -0.4109249319836895`, 0.6863212756169392`}
]]
Testing it:
clump3SlicesPlot3D[{
{texfunc1,Style[Log[Subscript[f, 1]], SingleLetterItalics -> False], "RedBlueTones"},
{texfunc2,Style[Log[Subscript[f, 2]], SingleLetterItalics -> False], "Rainbow"},
{texfunc3, Style[Log[Subscript[f, 3]], SingleLetterItalics -> False], "RedGreenSplit"}},
ImageTake[pdr, {1, 250}, {200, 400}], ImageSize -> 500, PlotLabel -> Style["Example", 20, Bold, FontFamily -> "Times New Roman", SingleLetterItalics -> False]]

Applying it to a list instead of functions can easily be done by using interpolation functions.
The general functionality is fine, but I think the displayed cross section areas are somewhat dim. I couldn't figure out a Lighting setting to make it more colorful.
Comments on this approach or alternative solutions are highly appreciated.
CenterPlanes
onSliceContourPlot3D
, withRegionFunction -> Function[{x, y, z}, Sqrt[x^2 + y^2 + z^2] < 1]
, and get two sperical plots? $\endgroup$ – Feyre Feb 7 '17 at 17:19