What does the Levite mean with his reference to “the house of the LORD”?

Judges 19:18 (ESV)

18 And he said to him, “We are passing from Bethlehem in Judah to the remote parts of the hill country of Ephraim, from which I come. I went to Bethlehem in Judah, and I am going to the house of the Lord, but no one has taken me into his house.

The Levite responds to the old man by saying that after a visit to Bethlehem he is going to the house of the Lord This follows the Hebrew text (בֵּ֤ית יְהוָה), as well the Targum and Latin Vulgate. But in the context, this reference seems odd. Though it is possible that the Levite’s journey could have included a visit to the shrine at Shiloh, in Judges that place is called the house of God (Judges 18:31). The Septuagint has and to my house, which is followed by most modern translations (e.g., NASB, RSV, NRSV). It is quite likely that there is a scribal error here, where an original ביתי, my house, was mistaken for an abbreviation of בית יהוה, house of the Lord.¹