Solarian Lunar weapon question

Hello people

I'm making a human Solarian and I like the Lunar weapon, since I am making a thematic char based on Count Dooku from Star Wars.
But I am a bit confused about the lunar weapon and it's dmg modifer. My question is does the lunar weapon count as a finesse weapon where you can add your dex bonus to dmg instead of str?
I have seen sources that say you can use dex as a dmg modifer and others that says it still uses str and the dex is only for the attack bonus. So what is it? I don't wanna cheat and add my dex to the dmg if is not allowed then I would just bite the bullet and have my flat +0 from str and +1 from my weapon crystal atm.

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Thanks in advance for the help.