The reach doesn’t really matter for climbing, just bend your arms more if you want to drop your shoulders down and forwards for your preferred pedalling posture. Your centre of gravity vs the rear tyre contact patch location is key to being able to climb without the front wheel lifting – move the saddle forwards and/or slide forwards on the saddle or stand up. I’m 2″ taller than you on a medium Soul with a 50mm stem, 711mm bars and a straight post with the saddle forwards of centre, and it climbs as well as I can (which isn’t bad on shorter climbs).
Regarding wider bars, they’re often a good thing but if they’re too wide for the gaps between the trees on your local trails they’re just frustrating – at 711mm mine require much more needle threading than they did at 670mm but they’re worth it for the handling with my long arms and relatively broad shoulders. 740mm+ would really annoy me round here but I’m sure I’d like them if I lived in the peak or the lakes or rode lots of trail centres.