I'm extremely late to this post but just wanted to say it's also possible to use the names given by the original creators.

Azure has both a name and a surname - Watage Mimikaki, Scarlet was originally Azari (I have no idea where the misinformation about his name being Akiri came from... a basic katakana read will prove this wrong) and Breeze was Tsurara. For Scarlet, it makes sense to keep his Japanese clan name Guren since it's backstory-wise important (so like, Azari Guren). When it comes to Breeze I think giving the characters' surnames of their creators is a nice nod towards their involvement (so Tsurara Abe would be the name). Tsurara is a name that could be a bit troublesome seeing how it means icicles in literal translation though...

Crimson and Midnight are a bit more difficult, but Kasugai Master and Apprentice by Onuma Rioka and Tsutome Benkaku by Yamauchi Chie are similar to them enough to get some names from there? Well, to each their own. But it's not like the 3 mains didn't have names before being adapted into the game.

But circling back to some points above, I don't think their English names are that weird. Scarlet, Midnight, Azure are all nice names. Their surnames are also proper sounding. With Breeze and Crimson it takes a bit longer to get used to but other than that *shrug*. Oh, thinking a bit more, isn't it funny how 3/5 characters were given color names hmm

The names you came up with are quite nice though!