The water in mid-June is not cold to swim it is fantastic 🙂
Renting bikes after exiting a ferry, then kayaking to the islet Melita is fine. You will find bicycle and sea kayak rentals of that on Mljet in Pomena, Polace, Govedjari near lake Veliko jezero
We are visiting Dubrovnik in June (2018) and would like to book a day trip to Mljet on the 28th June but none of the tour operators have that day available. Is there a specific reason for that? Can we simply take the ferry and enjoy the park without an organised tour?