I think there are plenty of ways to differentiate it from something like Firefly.
It's certainly an idea. I'm not super keen on that for Star Trek... i'm a bit more old school in my Trek and generally prefer to go more positive with it. I'm ok with like, rough-around-the-edges freighter captain who is a bit of a scoundrel, but a loveable scoundrel who is generally going to do the right thing, while maybe not doing it in the most above board methods.
If anything, rather than a Harry Mudd, I could do something more like a Rios, or like a Star Trek Han Solo.
Well, Harry Mudd can become a love-able character, haha. Even in Honor among thieves, the captain Kirk's character is also a love-able bard thief character. But, well. It's far from the usual Star Trek moral character.
I don't know, but why do I think about the old Miles O'Brien. You know, he has a family of four. But what if Keiko and his only son died in accident, and he becomes a very over protective father to Molly, his only surviving daughter. But because Molly join Starfleet and got assigned to a mining station in a remote area, Miles O'Brien decided to protect her by becoming a Freighter captain. And because he so familiar with Galaxy Class Starship, he modified an already mothballed Galaxy Class Starship to become a Dilithium Crystal Freighter. Claiming that the ship is freighter, when O'Brien modified it to become a warship, thanks to his experience as an Engineer. Just to protect his daughter from danger.