This explanation would be quite reasonable if Paul was writing to a synagogue of Jews but he was not, he was writing to a church comprised of Jews and gentiles. Gentiles would have little, if any, knowledge of what the Jews might have understood when Paul referred to "the Lord."
Although Paul was not writing to a synagogue of Jews he was writing to Jews, along with many Gentiles, who used to attend the synagogue in Corinth. After their conversion, they attended the church.
After these things, Paul departed from Athens and came to Corinth. And he reasoned in the synagogue every sabbath and persuaded the Jews and the Greeks. Acts 18: 1 & 4
Therefore, they would have had extensive knowledge of the Old Covenant. Gentiles also attended synagogues because they believed in and worshiped the Lord God of Israel. Paul also preached to God-fearing Jews and Gentiles in a synagogue in Antioch in Pisidia.
16 Then Paul stood up, and beckoning with his hand said, Men of Israel, and ye that fear God, give audience. 26Men and brethren, children of the stock of Abraham, and whosoever among you feareth God, to you is the word of this salvation sent. 42 And when the Jews were gone out of the synagogue, the Gentiles besought that these words might be preached to them the next sabbath. 48 And when the Gentiles heard this, they were glad and glorified the word of the Lord: and as many as were ordained to eternal life believed. Acts 13: 14 – 48.
These Gentiles also knew of the history of the Jews and prophesies of the Messiah. The same thing also happened in Iconium.
And it came to pass in Iconium, that they went both together into the synagogue of the Jews, and so spake, that a great multitude both of the Jews and also of the Greeks believed. Acts 14:1
Paul and Silas also taught the Jews and Gentiles about Jesus being the Christ from the Old Testament in Thessalonica.
Now when they had passed through Amphipolis and Apollonia, they came to Thessalonica, where was a synagogue of the Jews: And Paul, as his manner was, went in unto them, and three sabbath days reasoned with them out of the scriptures, Opening and alleging, that Christ must needs have suffered, and risen again from the dead; and that this Jesus, whom I preach unto you, is Christ. And some of them believed, and consorted with Paul and Silas, and of the devout Greeks a great multitude, and of the chief women not a few. Acts 17: 1 – 4
Since this was the custom of Paul why then should he not mention such things in a letter to a church that was brought into being through his ministry, as he already mentioned in 2 Corinthians 3? Selective readings out of the Law and the Prophets were read every Sabbath in every synagogue and Gentiles who attended the synagogue on a regular basis would have known quite a lot about Jewish beliefs. Jews were also well versed in their Holy Scriptures to the extent that some could also recite whole books from memory. I’m sure Jews would have had no difficulty in finding scriptures relating to the term ‘Spirit of the Lord’, just like Jesus did not have. The Jewish Scriptures did not have their books divided into chapter and verse which makes it all the more remarkable.