I have recently read that St. Nikoloas (yes, that St. Nik), a staunch anti-Arian, was involved in some sort of physical confrontation with either Arius himself or another of Arian at the Council of Nicaea. Various retellings of this story further claim that the Bishop received some form of censure from Constantine II as a result.
What, if any, actual historical evidence for this exists?
It's a hilarious story.
Okay, I can’t give you a conclusive answer without spending way, way more time on this than I can spare.
Here’s what I can tell you – almost no modern text in English cites a source for this beyond “later tradition” or “another account”, I can’t find anyone even giving a modern citation for it.
The 13th century Golden Legend of Jacobus de Voragine does mention that he was apparently at the council of Nicaea, but lists of bishops for that council do not generally include him. Jacobus’ text appears to be derivative from the vita of Nicholas written by John Diaconus of Naples from around 880 AD, itself a Latin adaption from the vita written by Methodios to Theodorum, which is dated early 9th century, and follows the work of Michael the Archimandrite.
For a fairly comprehensive list of literature on Nicholas, these two blog posts will help immensely": http://www.roger-pearse.com/weblog/2013/12/24/ancient-literary-sources-for-st-nicholas-of-myra/ and here: http://www.roger-pearse.com/weblog/2015/02/03/the-literary-development-of-the-life-of-st-nicholas-of-myra-santa-claus/
The next step would be to track down a copy of the Methodios' text and see if it appears there, but I confess that's beyond what I can offer you. Even then, it's several hundreds years later.