でしょうに is でしょう for inference, followed by the conjunctive particle に meaning "but". This に is the same に used in のに. You can think of でしょうに as a variant of のに but with an added nuance of inference. Its plain (non-polite) version is だろうに.
- 明日は雨が降るのに、出かけるの？ (informal)
Although (I know) it rains tomorrow, are you going out? - 明日は雨が降るだろうに、出かけるのか？ (blunt)
Although it's likely to rain tomorrow, are you going out? - 明日は雨が降るでしょうに、出かけるのですか？ (polite)
Although it's likely to rain tomorrow, are you going out?
まだ そんな心配する年じゃないでしょうに。
Although I think you're not old enough to worry about that(, why do you say that)? / But I think you're not old enough to worry about that!
There is no meaning of "obviously/当たり前" in your sentence. In the linked question, such a nuance comes mainly from そりゃあ (=それは). Its literal meaning is close to "As for that" or "When it comes to that", but by extension, it can work like an interjection that implies the feeling of "you know", "why not" or "it's only natural". For example:
- 「明日はサッカーを見るの？」「そりゃ見ますよ！」
"Are you watching soccer tomorrow?" "Of course, why not!" - 「なぜ嘘をついたの？」「そりゃあ…」
"Why did you lie?" "Um...(don't you understand?)" - Meaning of "そりゃあまぁ"
- かわりもしますよ what is the meaning of this も?