You probably won’t need any underexposure unless you’re looking for a very dark effect. Take a look at an 18% grey card – it’s about the tonality of heavy overcast. You may actually need to dial in a bit of positive exposure compensation if you want shadow detail.
This is snow not rain, but a low power flash pop can get you an interesting effect:
When I was younger I had a camera that I didn’t know how to use very well, so I could never turn off auto-flash. Every picture I remember taking that had falling rain had used the flash, and there’s a ton of silvery-white drops everywhere. If that’s the specific effect you’re looking for, awesome, but I didn’t think it was very interesting.