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I realized I didn't include the Flamesprayer in my list! You can get it by talking to Gilbert (the man at the window by the Central Yharnam lantern) after you defeat the Cleric Beast, but before you defeat Gascoigne. (It might technically be after you interact with the Great Bridge gate and before you interact with the Tomb of Oedon gate, I'm not sure what the actual triggers are.)

Oh yeah, can confirm, as long as you don't advance into the night realm you can get it from Gilbert in whatever order. I killed Gascoigne first, didn't touch the gate, talked to him, was just his old dialog. Killed Cleric Beast later and that's when I got the cutscene and the Flamesprayer.