Historically when the French Revolution happened and the monarchy abolished, nearby Kingdoms saw it as a huge deal. A constitutional monarchy with a puppet monarch might not care, but a monarch with any power would no doubt see Republics as a threat and not want that harmful ideology to spread to his nation and get him overthrown.
That might be true of the PRK (though its revolution is already twelve years old and has not spilled over into any other nation so far) but other republics, such as Avalia, Hambria, and Dong Wei have likely existed for a century or more and still many absolute monarchs are secure on their thrones. Not to argue they wouldn't be percieved as a threat, but likely less so since we have a different historical context. Communism/socialism would, however, be a huge negative modifier on most nations' lists.
Also, to clarify, the PRK is an athiest state but does not actively persecute believers of any religion.