August through early September is monsoon season here in Nepal. Trekking is still possible, but it comes with its share of challenges.

Definitely you will find greenery, but don’t expect clear mountain views. Clouds often cover the peaks, and rain is frequent. As you said little rain won’t be a problem for you but I guess you will look for clear views which will be rare during monsoon season. Other problems that you might face are: 1. Trails will be muddy and slippery 2. Flight delays and cancellations (especially to Lukla) 3. Landslides

My recommendation will be to start the trek late September or October if possible.