I'm not sure I agree. We do say 'more of a..' and 'less of a...' pretty often. (and I think 'just' means 'only' in the sentence, not 'fair') As an athlete I was less of a runner and more of a stumbler. Hm, it looks funny when you write it but it sounds natural. So I think this is mainly a spoken phrase. You could use most of the words in your sentence by putting the adjective after the article: It becomes less of just a happy time, but more of a time that you... That's correct in informal spoken english, but it wouldn't be written that way. It would be written more concisely: "It becomes no longer just a happy time, but more of a time that you..."

You can use "like" or "of" in this case. But I guess when we are talking naturally in America we do not use "of". When we write sentences like this they are kind of "smart" sentences. You only see them in books that are trying to sound like the old English writings, like Shakespeare. You can also use them when you want to sound sort of British. The upper class, or smart sounding, British English speakers will write and speak in a different way than in other English speaking places. And the old English comes from British English so that is why the writing styles are similar. You can also use these kinds of sentences any time if you have to write an important essay. Teachers usually like when your essays sound smart. So basically, sentences are not usually like this when talking or writing unless you are trying to sound smart or British or if you are writing an academic essay. We also don't use "just" in this way so much any more. Usually we say "I just did something" but the way it is used in your sentence means that you are calling something "fair". Also "unjust" is the opposite. It means something is unfair. So this sentence in everyday English: "It becomes less (of or like) a fair and happy time, and more (of or like) a time that you..." We can say "and" instead of "but" because "of" and "but" are what made the sentence sound old. Well I hope this helps. It is hard to explain this kind of thing. Maybe it is confusing. >_<