With the exception of 'Black Snow', I saw nothing that precluded this from taking place in the Prime universe.
As stated, there's the reference to consulting the Ferengi database. And there are some slight differences in the crew makeup -- Pitcairn as chief engineer rather than transporter chief, Garison as a lieutenant instead of a chief petty officer, Colt already serving as Pike's yeoman before "The Cage" -- though those could be chalked up to differences in interpretation.
And the whole Black Snow/cloaking device thing is a pretty big exception. "Balance of Terror" showed Spock treating cloaking tech as something new and largely theoretical, something he hadn't encountered before. True, ENT has established that some form of cloaking tech existed more than a century sooner, but that isn't irreconcilable with Spock having no direct, personal experience with the tech.
At the very least, TCoK differs from the Romulan story in Seven Deadly Sins
on the chronology and origins of Klingon cloaking technology.
I think the Ferengi database isn't an issue. The later shows were never consistent as to what was known about the Ferengi when.
Were Pitcairn and Garison ever given names in The Cage
? I've seen them listed in the transcripts as characters but never referred to by name on screen.
Colt does throw a wrench in things. Which is weird because she doesn't really serve a purpose that required the character to be Colt.
Haven't read Seven Deadly Sins
... haven't read much in the way of Trek books in a while.