Good post + repLet me just summarise this for you.
Jutsu of the same chakra nature negate eachover, as you stated in your post. Kirin is an offensive lightning tech, the 3rd raikages lightning armour isa defensive lightning tech. So they negate eachover.
Rasenshuriken isnt a lightning tech, also base rasengan has been shown to be stronger then chidori, so rasenshuriken vs a kirin, rasenshuriken would win.
However, rasenshuriken vs raikage, STILL had no effect. So Kirin would barely do anything to him.
Also the 3rd Raikages defense is the most solid in the entire narutoverse, he tanked an army of shinobi for an entire week until he ran out of chakra.. If a Rasenshuriken cant do the job, the only thing that can is his raiton finger.
Summary:
3rd Raikage's Lightning Armour --> Rasenshuriken --> Kirin
To bolded part:
I might be stating the obvious but since they both negate eachother, in this case its a pro towards 3rd and a con towards kirin.