In most cases, a closet is not required for room to be considered a bedroom. The city (or county) is much more likely to be concerned about safety issues. Things like egress windows and ceiling height are often issues.
FYI, many older homes do not have closets because they used armoires way back when and an armoire is not a part of the real estate (it is personal property unless it is attached to the house in a manner as to be permanent).
The space immediately outside the room is a hallway that only spans the width of the room with doors that open & close at either end and slider door into the room. Could this area be counted as an individual space with the bedroom? I don't want to use the word ensuite b/c there is not a bath attached.
We could put the access to the closet directly into the room, but it would take up the limited wall space b/c it is a small room.