Is this possible with Audacity?
Sorry, no - Audacity will import it into it’s own project format (with the previous MP3 compression) then you would need to export it as MP3 and add further MP3 compression.
Try MP3splt: https://sourceforge.net/projects/mp3splt/
Other MP3 splitters are available
I have tried mp3splt-gtk. Worst interface ever; can’t figure it out and it doesn’t even appear to have a waveform display, so impossible to finely tune where to cut.
mp3DirectCut is rubbish, too. Again, abysmal UI, not even sure how to create trim start and end points.
Why can’t Audacity be made to allow MP3 cuts without re-encoding?
Because it doesn’t edit the MP3 file - it reads it in an converts it to it’s own internal high quality format (32-bit float) for editing.
It’s just that Audacity is designed as a high-quality audio editor - not a simple cut.
You could always use Audacity to reda in your Mp3 and determine accurately your required cut point - and then use one of the many MP3 splitters.
If you can use FLAC (or WAV) you can avoid another generation of lossy compression.
Although you should avoid it whenever possible, it’s not necessarily the end of the world if you re-encode MP3 and sometimes you can’t avoid it. Use a bitrate at least as high as the original and listen to how it sounds. If it’ sounds OK, don’t worry about it. If it doesn’t sound OK you’ll can decide for yourself if you want to live with the degraded sound quality or “fight” with a different tool or choose a different compromise.
…I find MP3DirectCut a little awkward too, but I assume that’s just because I don’t use it often enough to be familiar with it.