Bava kama 83a, chachamim allow violating bechokosayhem lo saylaychu in taking an akum-style haircut.
Tosefta avodah zara 3 mentions the ability of a karov lemalchus to look at a mirror (an issur of lo yilbash in the times of chazal and rishonin… i don’t remember if it’s derabonon or deoraysoh)
Yerushalmi Kidushin 4:1, i made a mistake; it wasn’t a tanna, it was Nechemiah, they called him “hatasharsa”(the permitted one). Rashi in kidushin 69 says that this was necessary to show that nobody poisoned the king at his table, since they’re all drinking it too.
For psak in these issues, the beis yosef in YD 156 “velo” talks about it. I can get achronim and teshuvos if necessary.
But again, does this extend to attending a ceremony with religious elements and prayers? I don’t know. Does the British monarchy have any relevance to chazals melech? It’s a hard sell for me.