When loading a raster dataset into a geodatabase, you can specify a spatial reference that may be different from that of the raster dataset. However, projection-on-the-fly will not be applied to the raster because only the raw pixel data is used when loading into a geodatabase.
You could write a customized application that uses the Raster::SaveAs method to save the raster to another dataset. This will resample the original data and apply the transformation to the pixel data. Load the resulting output raster dataset, which will be in the desired projection, into the geodatabase.