The floor is integral with the side portion of the panel, so the answer to your question is no. The whole thing has to come out. First take off the upright back seat cushion (the curved L portion along the starboard gunwale). Then remove the seat base, hinges and all. Then the screws around the inside of the locker. Then cut the caulk where it meets the engine compartment deck. This is at the forward end of the section, it's only a 1 or 2 inch section. I didn't cut mine and it came up fine with a bit of effort. Once it's out you might have better access down into the area for the transom shower connections.
This picture shows you don't get direct access: (I have inboard engines so your engine room will look slightly different down there)
But this might be easier to reach than through the other panels in the locker. I'm guessing you could lay down on the cockpit floor and lean your head down into the space to gain access to the shower. I'd bring along a heavy mover's or utility blanket to cushion yourself against the deck or engine. Suffice to say you'd probably want to do this when the engines have been cold for quite a while. The only part that gets ridiculously hot is the exhaust manifold itself, but it makes for an easier job retrieving dropped tools when the engine is cold.
As for caulk, I was talking about the seal around the shore water panel 'hose bib'. The squarish plastic panel that has the hose connection in it. I don't think the locker itself can be removed (at least not without pulling off the whole engine room deck!)
I'll be headed down for hurricane prep tomorrow. Mainly to take down the bimini canvas, foredeck cushions and to double-up the lines. The tide tends to rise quite a bit during storms like this so I'm going to make the lines tighter away from the pier, so the boat doesn't float over and down onto it.