What kind of overshoes can I use with a large touring SPD cycling shoe such as the Giro Rumble VR?
You need overshoes specifically designed for MTB shoes. Note that most of them would say Road Compatible: Yes, MTB Compatible: Yes but they are typically optimised for road shoes. I have the Shimano S3000X NPU + overshoes which are marketed as MTB overshoes and they are great, although it looks like they are hard to find nowadays in reasonable sizes.