Something I've been thinking of for a while now: IRL, the coronation of the intended heir before the current ruler dies is a real thing, with it happening with Roman emperors and later adopted by European royalty, primarily the French but at least twice in England. It was a way to reduce the odds of a succession dispute, because the heir has been crowned already for a while. Really annoyed the nobility apparently, since it smoothed the transition of power and thus there was less opportunity to profit.
Since the Reach is French-inspired/flavoured, I would suggest looking at them for precedent of this. If there is, it's something we can present with historical basis. If there's not, we could still suggest it, though it'd be a harder sell.
We could incorporate this into our wedding. A ceremony confirming that we are the heir of the Iron Throne. I think that since we have a good relation with Oldtown,we could even rope the High Septon in taking part of the ceremony. It would be a great tool to exert legitimacy.