Is there something I'm missing here or is this just a luckfest?
Iirc, both Scarlet Gold Sword and Start of Heavenly Demise are composite attacks, so characters with both high DEF and MND shouldn't take much damage at all after a defensive buff. Always have a character like Tenshi, Meiling or Remilia in front, Remilia should be able to tank it even if she was built for ATK. Characters with one absurdly high defensive stat, like Patchouli with her astronomical MND, should also survive those attacks after getting a buff from Reimu or Yukari.
According to the wiki you need around 15.000 combined defenses to take 0 damage from Start of Heavenly Demise. That probably sounds high at first, but you should be able to keep a +50% defensive buff on Tenshi active without much trouble, reducing the combined DEF+MND requirement to ~10.000. Remilia shouldn't have trouble to keep herself close to 100% defensive buffs with her selfbuff, reducing the required DEF+MND even further.
You probably won't reach those combined values with Meiling, but it's not like you need to take 0 damage from it, just reduce it as much as possible.
Adjust your equipment! Items that boost both DEF and MND ,like the Robe of Twilight, are very useful here.
If you can you should also switch empty slots in before he gets a turn, in case you already lost characters in the battle.
I like to play an offensive strategy against Rinnosuke, then take a bit of time to buff up when he enters one of his weaker elemental forms (iirc the fire form is pretty harmless), and then just try to blast through his forms as fast as possible. Try to have someone with non elemental attacks buffed up for his final form, it's probably the most dangerous one. If he buffs himself with WSMR in this phase and then uses Start of Heavenly Demise it'll probably be over for you, so destroy the final form as soon as possible. Nitori is, as always, probably the best to end it as fast as possible here, since her Megawatt Linear Gun is just broken.