Let's look at history.
During TNG Sela and the Duras sisters collaborated. Now Sela officially controls the Romulan Empire, while Ja'rod Lursa's son has a lot of pull in the Klingon Empire.
Someone else who has pull in the Klingon Empire is Worf who remarked in his last canon appearance that "the Romulans fought with honour" (against Shinzon).
J'mpok rose to power on the platform of blowing up romulans due to Nero destroying a klingon fleet. He also subjected a major player in the Empire to discommendation for conspiring with the romulans, and reduced the man's House to ashes.
And though Worf may have given some grudging respect to the romulans, he does hate them. Not surprising since they murdered his family, and those allied with them killed his mate.
You can be a Star Fleet Klingon Captain because some klingons side with the Federation. It is only logical to assume that some Romulans would side with the Federation and Join star fleet. but that doesn't mean the romulans wouldn't have their own faction, as klingons serve in star fleet and they have their own faction. I support the Romulans being a playable faction for either side, and as someone has already stated, the Romulans are allies with Duras, so that is feasible, but if it sill bothers you klinks, take the hirogen as playable characters instead. Their thirst for violence would make sense in going to your side anyway.
The reason the Duras family fell out of power was precisely because of their alliance with the Romulans though. It soured all Klingons against them. Even in the STO timeline it suggests that J'mpok is weary of the Duras family. In fact, in game Klingons are noted to say exactly this.