What is 12am to 4am?

I've found this bar name called "cold dawn." But I think that was not a good translation. I do get the point that whoever named it made it "dawn" because its literal translation would be 'dawn', but doesn't it mean the time when the sun is already starting to rise? From its original letters, it's supposed to be between 2am and 4am usually. It's the time when it's still dark and anyone would feel it's in the middle of the night, and not clearly the morning. However, I also see that the English language calls this part of the night (technically) "morning" or "in the wee hour of the morning."

What would be a good translation for the bar's name in this context, or in other words, what do you refer to a time between 12am and 5am while you would feel it's in the middle of the night and you can get emotional so that it would be a good fit for a bar's name?

English has a word ("dawn") for the time when the sun rises (around 5-6 am, though it changes). I don't think English has a word for the 2d half of the night (midnight to 5 am), but "cold dawn" sounds like a good name for a bar.

what do you refer to a time between 12am and 5am while you would feel it's in the middle of the night and you can get emotional

Perhaps you should ask this in the Japanese forum, since it is a translation question. People who speak both languages would understand nuances that I struggle with (despite your clear English).

I have heard this time referred to as the "wee hours" or the "small hours" of the night.

"The wee hours bar". "The late night bar". "Late at night" is less specific. Depending on the context, it could be 11-12 or it could be 2-4 am.

If most bars close at 12, a bar that stays open later than 12 would be a "late night bar".

I'd use "in the still, small hours of the morning".

I agree with Ponyprof: wee hours or small hours would be common terms for that time.

I disagree with dojibear; I find the name "cold dawn" an unpleasant one for a bar, as (to me) it suggests drinking all night long until closing time (which here in New York is 4 AM) and then staggering home drunk and alone as the first light of morning starts to appear.