I wanted to create some maximally short and correct construction with any adjectives and those pronouns. I did some search engine research but there was nothing with "beautiful we", so what about the "we" pronoun? Also there was a song and some site articles with exactly "beautiful me" so it's probably correct?
"we" is a subjective pronoun, "me" is an objective pronoun.
We decided it should belong to me.
I decided it should belong to us.
Subjective pronouns don't take adjectives, and objective pronouns only rarely take adjectives.
I want to be the best me I can be.
The good me wants to eat healthily, but the bad me wants to eat junk food.
Note how even though "me" is in the subject, the verb has a third person singular conjugation.
So you can say "beautiful me" and "beautiful us", but not *"beautiful I" or *"beautiful we".
Your question makes no sense.
Phrases lacking verbs are not grammatically correct or incorrect although they may well be uncommon or puzzling.
In practice it would be possible to include both your examples in sentences in the right context. It all depends on how you construct your sentences.
He's not in love with clever you, he's in love with beautiful me.
Beautiful we are, my love, but wealthy we are not, so cut back on your spending.