@BlazingMagpie Yes, this is an interesting conflict.
I do think Koyanagi is being really selfish in wanting to cosplay for the wedding (though I'm not sure she meant for the entire ceremony, or after exchanging vows and goes in to guest entertainment.) as Naru said, this is also Kabakura's Wedding and his feelings should be considered as well.
However Kabakura's arguments are, in my opinion, weak sauce. Having an "image" to protect doesn't mean much to me. Although the manga has shown that Kabakura's image is used to whip the peons back to work, there is little to no indication that having his otaku side revealed would reduce his ability to make the office-slaves go back to work.
Obviously the way to settle this is talk things out and either make a compromise or one side relents (I'm betting on the former). Personally I think Koyanagi's cosplay example is too much for me. I don't think someone like Kabakura would want to get caught in a wedding ceremony with a masculine looking woman (though this might be projection on my part).
TL;DR Koyanagi is too being too selfish, Kabakura needs better arguments, both of them need to talk it out.