Looking at the opening verse you've quoted "The LORD appeared...", where the proper name of God, YHWH, is used, seems to set the context as being a Christophany. That it then goes on to say Abraham "saw three men", is not discounting that - it's quite common for angelic apparitions and Christophanies to be described in human terms, since they often initially appear to be normal men (I am reminded of Hebrews 13:2, "Do not forget to show hospitality to strangers, for by so doing some people have shown hospitality to angels without knowing it").
Then again, a little further on we read the proper name of God again:
13 Then the LORD said to Abraham, “Why did Sarah laugh and say, ‘Will I really have a child, now that I am old?’ 14 Is anything too hard for the LORD? I will return to you at the appointed time next year, and Sarah will have a son.”
Further indicating that God himself was actually present.
Now since God is Spirit and John's Gospel tells us:
John 1:18 18 No one has ever seen God, but the one and only Son, who is himself God and is in closest relationship with the Father, has made him known.
this indicates that it's the incarnate Son who makes God known within this creation; indeed Jesus is the image of the invisible God
Colossians 1:15 15 The Son is the image of the invisible God, the firstborn over all creation.
and Jesus himself says:
John 6:46 46 No one has seen the Father except the one who is from God; only he has seen the Father.
Getting back to our passage, we see:
16 When the men got up to leave, they looked down toward Sodom, and Abraham walked along with them to see them on their way. 17 Then the LORD said, “Shall I hide from Abraham what I am about to do? 18 Abraham will surely become a great and powerful nation, and all nations on earth will be blessed through him.[c] 19 For I have chosen him, so that he will direct his children and his household after him to keep the way of the LORD by doing what is right and just, so that the LORD will bring about for Abraham what he has promised him.”
Which very much seems like God himself discoursing with the other two angels who are with him.
Finally, we have the famous discourse between Abraham and God about Sodom and Gomorrah, in verses 27-33.
The only two interpretations other than a Christophany would be either (a) God the Father was manifest in this creation, or (b) angels are indistinguishable extensions of the presence of God, so much so that encounters with them are recorded with the proper name of God.
The sheer weight of evidence in this passage indicating the God himself was there, and the understanding that it is the Son who is the manifest presence of God in this word, leads me to conclude unequivocally that this was indeed a Christophany.