I tried to compress with the Fraunhofer MP3 codec, but I am only able to compress audio with only very low bitrates. How am I able to compress with higher bitrates?

Make sure that you do not use the Fraunhofer advanced codec, as that is the free version, that will only allow specific compression rates. To compress with higher bitrates, use the professional version of the codec, or use any of the supported encoder DLL or any of the supported external command line encoders.