Menachem Bluming discusses singing in the rainy sukkah:

There is only one mitzvah that you are exempt from if you are uncomfortable and that is dwelling in the sukkah. If the sound of the shofar is uncomfortable to you, sorry but tolerate it. If fasting 25 hours is uncomfortable to you, you are still required to fast on Yom Kippur. Why would you be exempt from dwelling in the sukkah if it raining?!

Sukkah is G-d’s embrace. He embraces us as we are without us doing anything religious. We eat, we drink, we talk to friends and that is sanctified by G-d when enveloped in the Diving Embrace of the Sukkah. In fact the verse “His right arm embraces me” refers to the sukkah. Unlike the days of awe when we pray and fast on Sukkot He embraces your life as you are, inviting your personal life to be imbued with sacredness.

An embrace is not an embrace if it makes you uncomfortable. Sukkot is too personal to obligate you if you are in agony.

Then again how can someone be in agony when embraced by G-d, even if it’s raining?

Chag sameach!

Mendel Menachem Bluming and Rabbi Moss and Chabad.org