would I be missing the meaning "beautiful ocean"/"beautiful sea" of ミ{mi}ミ{mi} if I used only katakana?
Yes. Whether you use hiragana (みみ), katakana (ミミ) or romaji (Mimi), no one would think you're referring to the ocean. If anything, people would imagine this word, which is not what you want to convey.
If you heard that Mimi meant "beautiful ocean" from someone, I think that person had the kanji 美海 in mind, which is one of the many combinations of kanji that can be assigned to the name "Mimi". 美海 indeed carries the meaning of "beautiful (=美) ocean (=海)". I would like you to read this Wikipedia article about the Japanese writing system, but the kanji 美海 is mandatory to convey the meaning of "beautiful ocean". If you write it in hiragana, katakana, or romaji, the intended meaning won't be conveyed.
Also note that 美海 is a girl's name, not an ordinary word defined in regular dictionaries. It doesn't make much sense outside person names. If you used the kanji 美海 in your logo, people would wonder if it's an anime character goods or something. What's worse, "Mimi (みみ)" is not the only reading of the kanji 美海; it can be read also as みう, みうみ or in other rare ways. A common workaround is to include both kanji and kana/romaji in your logo, as shown here and here.