I don't fly 135 (yet) but have my own airplane and sometimes fly in poor weather to maintain my instrument currency. Last Oct I went up on a day much like you describe - rain, mist, 800 ft ceiling plus a 60 kt wind at 3000 ft. Bumpy ride.
I was up at Frederick, MD shooting the ILS. Inside the OM on glideslope but the loc shows me drifting to the left. I turn to the right. Knowing that the wind is from the right this makes sense. However the loc needle shows me going faster to the left. I turn more to the right. Not long after the needle goes to full deflection and I execute a missed. This made no sense to me. It was almost like the needle had reverse sensing.
I went around and shot the approach again without a problem. When I got home I looked at my course on Flight Aware and it showed me right of the localizer turning right. I still have no idea what happened but it troubles me to this day.