I agree, digging up grass is extreme garden effort. Some potentially large areas! You can get mega-size boxes from furniture &/or appliance stores. Owner/operated stores will be easiest to work with, call or stop by & ask to speak to the warehouse manager.
You could do a row of cardboard between the shrubs & the fence to get started, &/or put cardboard in the major (black shapes) spaces, and newspaper in the red areas, if you are concerned about moisture penetration in the root zone of the shrubs. If the newspaper isn't quite strong enough in spots, you could spot-dig anything that breaks through later, or go over those areas with cardboard in the spring, after the first cardboard spots have decomposed.
Directly under the diameter of each shrub, I would put only mulch. That should drastically weaken the grass to the point where you can pull it out if it keeps trying to grow.
