Toadpool
I don't want to keep on about this but you are never thorough enough in your application of the rules.
First of all she (the mil controller) needed coordination as the traffic information that you issued became dead information and of no use to her once it was passed. She quited rightly needed this act of negiation between 2 or more controllers to agree a course of action as the ac had no mode C. Even though the FIS ac was 'not under control' you could have effected coordination by identifying the ac and obtaining compliance from the pilot that he/she would not fly above a certain altitude that meets the coordination criteria.
Lets not make this into a p1$$ing match but, had the ac been under RAS with unknown traffic with no mode C as a conflictor, she was spot on with her actions.
And if it is not an isolated incident have you tried to get your supervisors or watch manager to speak to the relevant unit and work towards understanding why the mil controllers insist on what you consider to be superfluous control actions. Without intruding on what seems to be inter-unit disagreement, the only way to resiolve it is to discuss it.