A buddy and I will be installing a second layer of composite shingles over the old ones already on my roof. Should we consider doing any of this in the rain?
It is quite possible to slide off a wet roof even if it's not that steep.
Shingles over shingles are not remotely close to vapor-tight, so they will dry eventually.
The problem with roofing in the rain especially this time of year is the temperature. The backings on the shingles stick better when it is hot. Even on a reroof the staples are doing some of the bonding but it is really the "glue" on the shingles that hold them flat. Not saying it won't stick but this is why it is best to roof in the summer. If it sticks well initially you won't have issues later down the line.
I would say no - do not install the shingles in the rain. I say that in particular because I highly discourage you from piling new shingles on top of old ones. And removing the old roofing is not something you want to be doing in the rain.
Any short cut that you get out of such an installation with new shingles on top of old ones is highly leveraged into a shortened life span for your new roofing material. New shingles will also look much better after a few years if they are installed on a properly flat surface.
Another aspect that gets missed with the short cut scheme is that you miss the ability to properly inspect the under sheathing for integrity. It may be that there are boards that need to be replaced due to dry rot of other types of damage.
And probably one of the most important considerations is that flashing, trim / drip edges and valleys are greatly compromised when trying to leverage the old stuff or cover it up.