Translation:Let's go and have lunch together!
In fact, I'd argue "let's go and have lunch" is different semantically. Yours is more accurate.
Is the together needed in Chinese? In English I would typically say "Let's go have lunch" with together implied by the us.
The meaning in English is still different though. Often, "together" is implied by context, but not always. It should be similar in Chinese.
If, at the end of a late shift at work, your manager says "Home time! Let's go get some dinner." you probably wouldn't interpret it as an invitation to join him.
the "s" in "let's" is an abbreviation of "us" so your sentence actually says "let us us go for lunch together" and is therefore not grammatically correct because you have repeated the "us".