On the PC version of the game, there are no real benefits to losing on 1-4. The DS version of the game allows you to unlock Prinny Commentary (where on the top screen of the DS, more dialogue is shown) and Pleinair for completing your first cycle. Since you can get Pleinair without needing to start a new cycle, and the Prinny Commentary is not available in the PC version, there's no benefits to be had there.
... However, the gameover music of the ending might be "special" in some sort of way - if it's a track you haven't heard yet, you'll unlock it. Like this, you could unlock the option to buy it in the music shop early. That said, you probably don't have the cash (100k HL) to buy any of the instrumental background tracks, nevermind any of the songs (10M? It's a lot of zeros).
I looked at an ending guide on gamefaqs (beware, SPOILERS!). It mentions that "it's got a vocal track you won't hear otherwise," so I guess that losing to Mid-Boss at stage 1-4 would be the easiest way to unlock that song.