Based on the Jekyll source code, it is not possible to retrieve the YAML front matter in a converter.
I see two solutions that could work depending on your situation.
Your file extension could be descriptive enough to provide the information you would have included in the front matter. It looks like the Converter plugin was designed to be this basic.
If modifying Jekyll is an option, you could change the Convertible.transform method to send the front matter to Converter.convert. The Converters that come with Jekyll would have to be modified as well. Fork it on GitHub and see if others like the idea. Here's where to start: https://github.com/mojombo/jekyll/blob/cb1a2d1818770ca5088818a73860198b8ccca27a/lib/jekyll/convertible.rb#L49
Good luck.