Your UV's look really messed up.
1) pepakura has problems with textures in certain formats like .kmz
2) sketchup is really bad at texturing.
I'd first suggest exporting to another format that pepakura can handle. .obj or .3ds
From the second image it looks like some of the UV's have been scaled, it's tiled, but some just look random.
However, when you look closer at the texture map repeated tiling, it looks like there isn't actually enough texture to cover the stabilizer as in the first image, so to me it looks like a texture applied to a decal placed over the model instead of being a texture applied to the stabilizer itself.
Any chance of seeing the actual texture, and a wireframe of the stabilizer in sketchup?
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