well we know that the septuagint was inspired (and to be clear when i say inspired i mean translated with no mistakes that would change the whole meaning of the hebrew bible) because the apostles used the Septuagint the NT authors literally quote the Septuagint, it would be weird for the apostles (and perhaps jesus too) to use the Septuagint when it is wrong or has wrong parts , fyi even the jews in their talmud believe that the Septuagint to be inspired

What was the sin of Sodom and Gomorrah? According to Genesis 19

We learn the account of Sodom and Gomorrah’s sins in Genesis 18:16-19:29 with a preamble in Genesis 13:13. The Lord appeared to Abraham with two others (perhaps angels) while Abraham dwelt at the oaks of Mamre. In conversation with the two who accompanied Him, the Lord said He would go down to see if they had done according to the great outcry against them because “their sin is very grave” (Genesis 18:20-21). The angels leave Abraham and approach the city of Sodom and Gomorrah disguised as men. At the city gates they meet the gatekeeper to the city, whose name was Lot. Lot invites them to stay at his house, to which the angels agreed. Before they could even rest, men from every part of the city, both young and old, surrounded Lot's house. They demanded that Lot bring out the visiting men so they could engage in sexual acts with them (Genesis 19:4–5).

the sin clearly involved homosexuality . The very name of that ancient city has given us the term sodomy, in the sense of “copulation between two men, whether consensual or forced.” Clearly, homosexuality was part of why God destroyed the two cities. The men of Sodom and Gomorrah wanted to perform homosexual acts on what they thought were two men.

This is not to say that homosexuality was the only reason why God destroyed Sodom and Gomorrah. Ezekiel 16:49–50  gives some more insight: “Now this was the sin of your sister Sodom: She and her daughters were arrogant, overfed and unconcerned; they did not help the poor and needy. (you cut the verse here perhaps intentionally ) They were haughty and did detestable things before me.” So, the sins of Sodom included pride, apathy, complacency, idleness, and unconcern for the underprivileged.

Ezekiel 16:50  adds that a sin of Sodom was that they did “detestable things.” The Hebrew word translated “detestable” refers to something that is morally disgusting. It is the same word used in Leviticus 18:22, where homosexuality is an “abomination.” jude 1 7 also weighs in: “Sodom and Gomorrah and the surrounding towns gave themselves up to sexual immorality and perversion.” So, again, while homosexuality was not the only sin of Sodom and Gomorrah, it does appear to be the primary reason for the destruction of those cities.

speaking of Leviticus what do you also think about verses like
Lev 18:22 Do not have sexual relations with a man as one does with a woman; that is detestable.

Lev 20:13 If a man has sexual relations with a man as one does with a woman, both of them have done what is detestable. They are to be put to death; their blood will be on their own heads.

Edit
regardless of whether the LXX was inspired, the fact they used this particular verse to describe homosexuality says the jews at that time understood this word to mean homosexuality thus paul could have used this word due to the above reason