If I remember well, RMSS requires pre-declaration of actions.
As such, the character must have pre-declared Tumbling Evasion (given that they will not attack this round). IMHO, there is no reason to prevent the character from doing it to avoid the incoming attack, regardless of initiative.
More generally, I think the RM initiative is broken when including defensive skills, unless it is used only to resolve the relative order of conflicting actions: it can be useful to determine who resolves their attack first when two opponents attack each other, but cannot be used to determine who "goes first" between an attack and the corresponding defensive action.
Furthermore, conflicting active manoeuvers (e.g. race to see if the character can snatch the McGuffin away before the evil minion can grab it to complete the ritual) are best resolved by opposed skill rolls rather than initiative rolls. Globally, initiative is not really useful, because it is less important than what makes sense in the fiction.