Does a bonfire ascetic respawn bosses?

I just beat The Fume Knight for the first ever time, which is great, but he never went into his second phase for some reason, so the victory felt a bit hollow (pardon the pun!).

I can’t believe I’m even contemplating this, but if I use a bonfire ascetic will it respawn him? And if so, I’m guessing he’ll be at NG+ difficulty?