Currently, the Esri FGDB API does NOT support rasters.
Here is the description of the File GDB API from Esri: File Geodatabase API details
An ArcGIS Idea has been set up with this exact request: File Geodatabase API support for rasters
According to a comment on that site, plans to support Rasters in the API are not even on the radar. I would encourage you to go and add your vote to the Idea, and spread the word around to others to do the same exact thing.
A search on the SAFE software website shows something interesting though. They do have support for the Raster format in the File GDB. Take a look at this page: Comparison of FME readers and writers for SDE and Geodatabase
This page may also help: Notes on FME and Esri Versions and Compatibility
The name of the reader/writer you should be looking for is: GEODATABASE_FILE_RASTER_DATASET
This must be using different driver technology than the FGDB API. This would make sense given that SAFE is an Esri business partner and thus would want to be able to ETL all of the formats that Esri has available.
The driving question is probably whether or not you are able to install ArcGIS onto the workstation that FME is installed on. Doing this will allow you to access the ArcObjects based driver that allows Raster access in the File GDB, as opposed to being limited to the FGDB API based driver.