The interface to Trimble Connect from Business Center - HCE is typically going to be a LandXML file or the VCE file (but that actually uses ESRI Shape Files and is not really a VCE File at all (smoke and mirrors by the Geospatial Team). Neither of these file formats support Textures at all.
If you output a Sketchup SKP File it sends the surface out with the Surface Colors as defined by the MSI Library but not the Textures (because you have to match the Textures in BC with the equivalent texture in Sketchup I believe and that has to be mapped correctly (I am not clear whether that is possible or not today as it doesnt look like the Textures come through with the SKP file data.
You can Texture a BC-HCE Surface in Sketchup and then save it to Connect - If I open the SKP File on Connect using the Sketchup Web Editor it looks the same as it did in Sketchup. If I open it in the Connect Viewer the colors match those of the Textures, but the Texturing scale is either very small or the color only is picked up as the Texture itself is not discernable. Since I am not so heavily involved with SiteVision - I am not sure what it is doing with the data right now - but if it opens and Reads the SKP file natively then the Texturing is there and it should be something that can be picked up by the software - however if they are converting the SKP into something else - or if they are using the Connect Formats in the system then it is likely you will get the colors as per Sketchup is showing below or Connect is showing (2nd Image)
I will share this link with the SiteVisionTeam for their input as they will know better what formats they are using etc.