Beechcraft 1900 pilot reported missing approach due to lateral deviation prior to landing. Pilot missed approach and diverted.

Narrative

Normal flight until final approach. There was a rolling fog on the airport surface with shifting levels of visibility (1600 RVR to 2200 RVR) at 750' AGL and below. Clear visibility above the fog on the approach. I intercepted the glide slope and started down on final. For some reason I expected to see the approach light system sooner than the approach minimums (which in hindsight was an incorrect expectation). I transitioned from looking at my instruments to starting to look up and out to see if I could find the runway visual cues a little too early. That led to me starting to drift a little left then right as I corrected the plane based on the instruments and then looked outside again.Approaching the last 200' of the glideslope descent; I got a little too far right of the ILS centerline; likely 3/4 scale deflection and just as I was about to called missed approach the Tower told me to go missed as well. In a subsequent call with the Tower they told me I had triggered a warning in their system that I might be drifting too far right. I executed a missed approach/go around and ultimately ended up diverting to ZZZ to land since the visibility was getting worse at ZZZ1.The critique I gave myself after the flight is that I should have waited until I was about to reach my decision altitude before looking outside for visual references. I would have maintained the ILS lateral path with greater precision. Allowing myself to start looking outside too early introduced some small but oscillating corrections that led to decreasing safety margins for a moment about 100' above the decision altitude of a challenging ILS approach.Also; the clear visibility on 95% of the approach with the last 5% in fog was a unique approach challenge I haven't had much opportunity to experience prior. Usually it's the been the reverse for me where I'm descending through low visibility/clouds and breaking out into greater visibility before reaching my decision altitude.