An air taxi pilot reported a rough running engine and diversion to a nearby airport.

Narrative

There was a First Officer riding with me to ZZZ1 so I had him sit up front with me and plugged in. Runup and takeoff was normal. On climb out at about 1500 ft. I started to reduce to climb power and the plane started to yaw towards the left and the engine started to run rough. I put the power back to full and the engines were still running rough. I looked at the fuel flow and it was normal. I told ZZZ Approach that I wanted to return to ZZZ. They gave me a heading for the approach back into ZZZ. It was low IFR in ZZZ and the approach lights were Notamed out for the runway; so I asked ZZZ what the weather was like there and they said it was marginal. I elected to go to ZZZ2 because of the weather and the services that they offer. They gave me a heading for the ILS XX. I gave the QRH to the First Officer to go over the different checklist that we would need/use if the engine failed during the different phases. We leveled off at 3000 ft. and about half way to ZZZ I noticed the left fuel flow was 245 and the right was about 215; while at full power; so I reduced the mixture to 200PPH and it started to run better. Once we reached the initial approach we were cleared for the ILS and shortly after were able to see the runway. About 5 miles from ZZZZZ I started work the power down to in range power. The roughness of the engines went away and it ended up being a normal approach and landing. I taxied to the FBO with no problem.