I don’t have too much knowledge of historical context around it, but having grown up for a large part of my life in a French environment (France !) I’m almost positive the word majesté (literally « majesty ») is the best word to associate with « illustre », in terms of frequency of use.
Though the Monégasque princely family seems to be using a similar phrase, your idea of using son altesse illustrissime sounds mostly like a humorous exaggeration to a mere mortal, while son altesse illustre could probably work if you instead said « illustre altesse » (French can also have adjectives before the noun, when the sentence is respectful or as a way to make it elegant). However, I think « illustre majesté » is best because the adjective contains 2 syllables, versus 3 for the noun, allowing it not to be overshadowed. But perhaps I’m missing a semantical difference that leaves « altesse » as being a better word choice.
Also, « Son » and « Votre » can be used interchangeably (unsure which is most respectful though).