You asked for specifics on sea creatures big enough to hold Jonah? Sperm whale, whale shark, and white shark. I hear that these have all been found carrying whole creatures big or bigger than a man. Additionally, other creatures could qualify as well that may not be as common today (dinosaurs) and these creatures may also come in larger than normal sizes.
How could he breathe? A whale's stomach contains some air, and there are multiple possible answers. One of which you already acknowledged, that of having the fish come to the surface a few times. Another that I already mentioned was that of slowing Jonah's metabolism, which hardly seems supernatural seeing that it's a technique known to Bhuddist monks today. So what if Jonah wasn't Buddhist? Considering that he's already steering the fish, it's not that hard for God to do it for him.
Stomach acid and bends? These are not our concern, because God wasn't trying to give Jonah a ride in comfort. He just needed to come out the other end of the experience alive. Regardless, by this point you already granted that "God prepared a great fish" for the purpose of swallowing Jonah. If one is preparing the fish to swallow him on command, it's a small bit extra to make sure that it's stomach acid is dilute enough to not dissolve the target (sea water anyone?)
Jonah and the whale is one of those examples for which it doesn't require much imagination for feasibility. Matter and energy remain constant and the minimum required supernatural interference is instructing the trained whale (large sea creature.) You might as well protest the storm that preceded the whole encounter, and claim that "it's not scientific that God could create a storm!"
Let's put this in perspective. You protest that an intelligent being able to create life from non-life, man from rock, is unscientific. I protest that an unintelligent accident creating life from non-life, man from rock, is unscientific. Only one of these has been observed to have happened. I think my scenario has a significant head start on yours.
I could certainly dispute it. To quote Captain America, "There's only one God, and He doesn't dress like THAT." And besides, I was here last Friday at noon, so yes, I can disprove it through witness testimony.
Should we include supernatural under science of study? It depends what you mean by that. Spirits are known for being unpredictable, and I don't recommend trying to summon them or dabbling with the occult. So as a general answer, no. However, that is a far stretch from being so blind as to deny that supernatural does exist.