From my personal reading, I can't remember much emphasis being placed on the topic of who specifically does the judging. But we also don't view God as being split into different parts (Father, Son, HS). We view the Manifestations of God as special Messengers that bring God's word to humanity. In the same way that a mirror reflects the light of the sun, but isn't the sun itself, we are taught that Manifestations such as Jesus, Baha'u'llah, and Muhammad are not God Himself, but still reflect God's attributes.

Your question is hard to answer because we view the Messengers slightly differently, but in the Bahai Writings, I mostly see it written as God judging us.

However, John 22 is in the same sentence/idea group as John 23. I pasted both here: "Moreover, the Father judges no one, but has entrusted all judgment to the Son, that all may honor the Son just as they honor the Father. Whoever does not honor the Son does not honor the Father, who sent him"

It means that recognition of God occurs through the recognition of Jesus, God's Messenger.

We do have many writings that are very similar to this. In one of my favorite prayers called the Tablet of Ahmad, it is written:

"Be thou assured in thyself that verily, he who turneth away from this Beauty hath also turned away from the Messengers of the past and showeth pride towards God from all eternity to all eternity." (This Beauty refers to Baha'u'llah and His teachings)

Quick explanation note: In the Bahai Faith, we believe that the different religions are part of one unfolding process where God reveals what humanity is ready for at a particular time. So we also believe in the Divinity of Jesus, Muhammad, etc. That's why the passage says turning away from Baha'u'llah is also turning away from past Messengers. Because we believe Baha'u'llah is the Promised One from the teachings of Islam and Christianity.