I would not swap out "The LORD" for "Yahweh" for these reasons. The Greek OT (the Septuagint, often abbreviated "LXX") consistently renders the Greek Kyrios for the Tetragrammaton (YHWH). Kyrios, as you probably know, is the Greek word for "Lord."
The LXX was the Bible for the earliest Christians. When the apostles quoted the OT almost invariably they used the LXX. It had authority. So if the LXX uses "Kyrios" for YHWH it makes sense for us to translate LORD when we want to render YHWH in English. We are doing in English what the translators of the LXX did in Greek.
Furthermore, the New Testament laid a close connection, even an identity, between Christ and YHWH by calling him Kyrios. The LXX is an important link for us between the OT and the NT. Christ assumed unto himself the names of YHWH, and that assumption very much hinges on the name "Kyrios." In English this link between the Lord Jesus Christ of the NT and the LORD of the OT is made clear in the name Lord/LORD, and the LXX is the link. If we did the swap-out, we'd lose that which the LXX made clear to us.
Of course I realize that I am justifying the LXX. Perhaps the case can be made that the LXX translators started us down a wrong path by rendering YHWH as Kyrios, but they did what they did, and I think the link of the name Kyrios for YHWH in the OT and Jesus in the NT is a wonderful gift that exalts Jesus as the Son of God, as God himself.