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, you are required to 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 a hug 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