The issue here is that the fme_basename is an attribute of the raster mosaic feature as a whole, and not the individual cell.
@takashi could probably whip up a python solution to produce a source matrix, but working just with transformers I would consider experimenting with the following.
Force all rasters to the same extent and resolution. (If not already)
RasterBandCombiner to create a single raster such that raster 1 => band 0, raster 2=>band 1 etc.
RasterExpressionEvaluator to create a band where the pixel value contains the band number that has the highest value
Associate that band number to the fme_basename using the RasterPalette transformers.
Hi @jayec
Expanding on the answer from @jdh, I use a RasterExpressionEvaluator to compare a copy of the mosaic raster to the source images, setting the pixel to an index value corresponding to the raster source. The index rasters are mosaicked together to form a single index raster, which output along with the original mosaic, and a CSV of the source filenames and their index values. A string palette output might be better, but I don't know of a format that we can write that to.
I have attached the workspace: calcmosaicsources.fmw
Much thanks to @jdh and @DaveAtSafe for your assistance. I utilized both of your suggestions and it worked out perfectly. A string palette wasn't needed as I essentially linked the CSV to the index raster and from there was able to combine that with the overall mosaic. Thanks again!