Translation:Let's go to karaoke tomorrow.
No native speaker would reasonably use "Let us" here because in modern contexts, it is never used in equal social standing. "Let us pray", "Let us go see what all the fuss is about" or other rhetorical constructs occasionally are used but the underlying assumption is that the person speaking is higher status than the listener and would never expect them to say no. It's similar to Wouldn't...? ->Would not...? style questions where the formal tone brooks no argument.
That's one way to interpret it, certainly, but I disagree that it inherently implies a difference in social status. If we really wanted to, we could imagine this being someone of high social status, or pretending to be, speaking to another of equal rank. Register is certainly a factor, but that needn't be bound to rank.
I've already highlighted several possible uses and these all seem reasonable to me, but then perhaps I am not such a reasonable native speaker (and I do assure that I am a native speaker). I've no issue with using it, nor do I with "would not" — as per your example — if the situation demands such; be it for emphasis, authority, or what have you. Words are tools to be used to express oneself.
We can see in the translated sentence that "let's" is shown as the translation, as it probably should be, so as not encourage any potentialy costly social mishaps. I'm fairly certain this has been the case since I made my initial comment, whether it has been changed in the exercise or not.
My intial comment hinted at essentially everything anyone has said against "let us" here, so I'm a mite surprised there has been so much said. I stand by the fact that it is uncommon, and may even prove downright awkward at times, but it is not incorrect. If you choose not to use it in "common speech" that's on you. If I look like a pompous fool for doing so, so be it.
No, I did understand you.
In my experienve at least, it is used "to some extent", even if that is only to intentionally sound old-fashioned with humerous intent. More often perhaps it is to be sarcastic, pointed, scathing, derisive, or just emphatic, which decoupling the contraction tends to do. Common? Maybe not, but certainly not never used.
I'm sure in many cases though it would sound more like asking for permission on behalf of more than one person rather than making a suggestion.
The important thing is NOT what we say in English - this isn't an English learning exercise, it's a Japanese one. If non native English speakers have entered "let us" instead of let's, they should be given the point. It's archaic English, sure, but that doesn't actually matter because it does not substantially affect your understanding of the Japanese.
As far as I can tell, カラオケ is a noun in Japanese, but is like many other nouns used with をする to turn it into an activity. So, カラオケをしましょう (let's do karaoke) would be a way of using the word like a verb. カラオケに行きましょう (Let's go to Karaoke) isn't wrong either, as it's used in the same way as we'd say "Let's go to the Arcade." In Japanese, if you want to specify the establishment you want to do kararoke in, you'd say カラオケバー or カラオケ屋、for Karaoke bar and establishment respectively.
I know it says "ni ikimashou", so technically "karaoke" should be treated as a place rather than as an activity, but "Let's go do karaoke tomorrow" is what I put as it seems most natural. Either way, "to karaoke" does not appear to be correct (as "karaoke" is not being used as a proper noun). It should be "to a karaoke (bar) or "to the karaoke (bar)".
I see what you're getting at, because we say "go to the zoo" or "go to the park", but it's not about being a proper noun.
"go to school" - school is not a proper noun
"go to work" - work is not a proper noun
"go to dinner" - dinner is not a proper noun
"go to karaoke" - completely natural to me and and a phrase I use often
I'm interested to know where you and the others who find this phrase unnatural are from. Maybe we can narrow down the region/dialect.