Chris Warner and I planned to attempt Amotsang during August and September. In 2003 a French expedition led by Paulo Grobel reconnoitered the south and west sides, finding no feasible route. Moving to the north flank, they decided the best approach would be directly from the east, but made no attempt on the mountain. Using maps and Google Earth, I felt an approach was possible via a col to the north of Pokharkan, dropping from the col west into a valley east of Amotsang, before climbing into a glacial cirque north of the peak. However we only got to see the second part of this route once, for an hour, due to terrible visibility, and that was late in the trip. It is certainly feasible for a small team, but porters would help.
We eventually attempted a new line on the north face of Pokharkan, from a high camp on the north col (ca 5,400m), but only reached 5,900m. I stopped due to poor visibility and minor avalanche danger; there was a lot of light rain, very cloudy up high, and hard to see where to go. Some past reports have indicated August and September to be fine for this area, due to the rain shadow formed by the Annapurnas and Chulus. This was not so in our case.
April and May might be too early, with snow blocking approaches, and October is often very windy up high, as in Mustang. Late May through June or late September is probably best.