First, let me say I for one am not married to the "Royal" desig - But "Canadian Navy" would rally the old Royal bunch as well as the more modern and "republican" younger generation behind something we can all ...well, get behind!
As for the SSi, the difficulty in implementation will be self induced if we insist on the goddamned centralized control that seems to pervade any "administrative" task. Otherwise, it could happen fairly easily.
To start with you pay a visit to a little appreciated Directorate in Ottawa: DHistory. Every year, this Directorate receives from each ship the documents that junior O's and Mars IV's, under the whip of the Nav Os, have come to hate: The Ship's Log. This Directorate could easily, from these documents, generate an approved calendar of days each ship in the navy was considered "at sea" for the last 30 years. I am sure the professional historians that are custodians there would love the task, which would be right up their alley.
Then, each unit individually, using the approved calendar and the personnel files showing postings, would review each seaman's file and "grant" her the days at sea that correspond to her posting periods to any ship on the calendar. This grant could be in the form of a memo signed by the CO put in the pers file. To avoid pointless discussion, we would ignore for the original implementation, details such as leave period or medical absences, etc., as well as ignore temporary transfers for a few days here and there. The figures generated may not be perfect, but they would be generally correct and rapid implementation would be the benefit derived from waxing over the details.
For the future, any individual on a ship would have his days at sea recorded in his pers file, and for greater certainty, indicated in his PER. With most units having between 2-3 hundred individuals, such a review could probably be carried out within about a year.
Here is a small point to ponder: The SSI is not a trade designator, special qualification badge, medal or ribbon. It is an indicator of sea time, akin to the old years of service badges. Most navies in the world have some form of service badges, stripes, pins, etc. to recognize and affirm such time in the sea service for their seaman - however, I am not aware of any navy, and I am certain no "British" style navy, that has such a "badge" system for the officers. Might it not sound a little goofy for officers to explain what that thing is on our uniforms when we meet our colleagues of other nations?