Wearing a cloak to hide identity, does not attempt to use it to cover up. Also, how thin was that outfit to become transparent with a little water? Me thinks she is walking around town in what amounts to a nightgown.
Last I read the WN, the crown prince's death is treated as a background mystery. The plot's main focus is on the power struggle and MC's attempts to help his sibling win (MC considers the death mystery a lower-level priority over ensuring his immediate family's survival.)
i think this "Marie" is a spy sent by one of the princes and she seem so suspicious because of her expressionless face as
if she expressed her feelings she would kill them as is a spy who despises the prince Arnold and wants to kill him and our MC