You will definitely need an axe to get to exum ridge still. The lower ridge didn't seem to warrant an excessive rack, but I did it after a 2 month trip in the Winds, so I was in silly shape. I'd say a set of nuts and singles to #3. There are good ledges for belays the whole way. Finding the start of the lower ridge in the dark is a beast, so good advice about scouting. If you can't find it, you can do a couple of pitches on the Petzoldt ridge and traverse back to exum.
The thunderstorm that hit recently was no standard afternoon shower. I think the accident occurred around 11 am? Another reason to use everything you can to your advantage and move quickly, especially on exum. Those guys were on the O-S, the descent route, and they still got hit. On exum you would have to continue over the top after an incident like this. At least to the level of the v-pitch, and that escape is not obvious.