Is the expression "(someone is a) bad loser" valid?
If it is valid, is it equal to "sore loser", or does it have a different meaning and/or use?
"Sore loser" is a more common idiom to describe someone who acts badly when they don't win. But "bad loser" can also be used to describe this, and is transparent in meaning. I have heard "bad loser" from time to time, but "sore loser" is much more common, simply because it is an idiom that has been around a long time.
Sore loser is a more common expression than bad loser.
You hear and say things like "this fish tastes bad." or "This is a bad idea."
So, bad loser would only make sense if the character was a bad person. I don't believe "bad loser" makes much sense in your case. I would even go so far as to say you're better off using "poor loser." While it may technically be okay to write and use "bad" loser, it doesn't make much sense.
I once used "bad loser" in a story I was writing and my professor goes "What's bad about him being a loser? What's the backstory, is he a bad person?"
I forget the word he used, but he essentially said something like: "Pork fried rice is bad, a failing grade is bad, being a pervert is bad. Being a loser cannot be defined as 'bad' and is more subjective than anything." Again, this was back when I was in college, so excuse the lack of accuracy, he explained it way better than myself.
The dictionary defines bad as: of poor quality or a low standard.
You're kind of restating the statement twice.
In technical translation, it's like you're saying "He's a bad poor loser." Loser already establishes the fact that, well... he's a loser.
The dictionary defines sore as: extremely; severely. (of a part of one's body) painful or aching.
So in technical translation: "He's painfully stupid; a severe loser."
Essentially, my thought is that you should stick with poor or sore. It's not that "bad loser" cannot be interpreted to mean sore loser, it's just that people don't always look at that the same way.
As an English-focused major in college, I learned that what may be wrong in one place may just sound different, but will not necessarily be wrong somewhere else.
To answer your question: it's valid to say "bad" loser but it may not make sense to a lot of people as it's not common everywhere, definitely not as common as "sore" loser.