Can I get Handbrake to apply additional filters such as brightness/contrast?
Like you suggested yourself you might want to use FFmpeg instead. It can utilize x264 as well which is the library that Handbrake is using for encoding. While x264 itself can do some very limited color correction via color space conversion (which can be used in the Handbrake CLI) I wouldn't recommend it if you want advanced manipulation of the video look.

FFmpeg isn't hard to use and not all too different from the Handbrake CLI.
The advantage you have with FFmpeg is a lot more formats you can output to and read from and you have the option to use many advanced filters on your video aswell as audio:
http://ffmpeg.org/ffmpeg-filters.html#Video-Filters

It's a very powerful tool for automated video editing and conversion.
The downside to FFmpeg is that you don't have all that many excellent presets but you should be able to use the x264 presets from Handbrake aswell.
https://sites.google.com/site/linuxencoding/x264-ffmpeg-mapping