As a DM, how would/did you handle or describe this situation? How easy or difficult should noticing and taking this silver baton be? Is this a hidden treasure?
I think it's good that you included the previous room, because it provides context.
My description, based on "This was once the domocile of the major domo of the castle, but is stripped of everything but broken and ruined furnishings now":
"This room was appears to have once been a grand bedroom or audience chamber. The walls are lined with peeling and faded paint, and a row of sconces to either side. One black cresset lies neglected on the floor against the wall. In the middle of the room you enter you see a smashed wooden chest, its fittings now rusted red and pitted with age. A large armoire lies against the left wall, similarly rudely splintered. On the opposite wall is a fireplace, now choked with rubble and bricks. The remains of a wooden railing and gate lie toppled halfway across the room. Beyond is the bottom frame of a large bed, enclosing a rotten pile of down and mixed linen. The dust is heavy everywhere and scraps of cloths lie rotting on the floor."
As a Player, how would/did you handle or react to this situation?
If I was under time pressure, this would likely prompt me to search the walls for hidden doors and then move on. The description conveys to me that this room has been picked over already and little of value is likely to be found here. Indeed, that's true, as the 30 gp baton is barely worth taking compared to its weight in many circumstances, even if you could find it. Also, even if I was inclined to make a search, if I'd searched several similar rooms and found nothing, I might hesistate to make an exhausitive search again. And generally speaking, rooms like this are seldom worth searching, as its 50/50 that you find something you'd rather not find like a patch of yellow mold or some other dungeon hazard.
If I wasn't under time pressure, this would likely be a 'we search everything in the room'/'take 20' situation and hope the DM let's me hand wave the details of the search - clueing me into the fact that the room is empty of danger.
How easy or difficult should noticing and taking this silver baton be? Is this a hidden treasure?
In 3e terms, finding the silver baton requires one of two courses of action.
A proposition that you investigate the cresset, and a DC 15 Appraisal check or a) recognition by the player that if the cresset was iron it would be rusted red like the chest furnishings, not black or b) an attempt to remove the corrosion from the baton with a mild acid or vigorous rubbing resulting in a much easier appraisal check (DC 5 or something) when the silver color is revealed.
A proposition to search everything in the room, and an accompanying DC 15 Appraisal check.
Since the cresset is apparant from my casual description of the contents, its a fairly trivial search check so I haven't included it here. With more time to think, I probably could add 1d6 peices of nearly worthless junk to be found on trivial searches and then mix the cresset in amongst them. If the players were very experienced, I might justify a DC 20 Appraisal check on the grounds the clearly someone has searched this room before and missed it.
I consider this to be in context hidden treasure. This is a good basic treasure finding problem for an inexperienced adventurer.