If we want to keep up with the times, perhaps the new timeline's Federation-Klingon relationship should be more akin to modern Sino-American geopolitics. It would be a return to a more totalitarian-styling of the Klingons (a la TOS), rather than "imperial samurai Viking bikers."
Remember, the Klingons probably lost 15-20% of their fleet in action against the Narada in 2258, so they're also wounded and probably angry with the Star Empire (possibly not accepting any explanation by the Romulan government as to Nero's status within the empire, whose response was probably something like, "Who?").
It possible that, while they're focused more on the Romulans (conveniently keeping them from taking advantage of the sudden perceived Federation weakness), they can only engage in a proxy-war by funding/supporting a belligerent smaller power (akin to the DPRK) into harassing Federation assets. (As I've said before, I would love to see something like this played out on Axanar.)
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