When I planned these beds I spaced them far enough apart to get my lawn tractor in-between them, but getting between them and the fence involved my weed whacker. As anyone with a fence has found out, maintaining the grass at the base of a fence is a pain.
I'm 3/4 of the way done with the edging. It's 10" tall with something like 6" or 7" of it buried. It does a good job of keeping grass out of our other beds, so I'm sure it will do a good job here. The downside is the most effective way of installing it is to trench first, put the edging in, and then refill the trench. If you try to use one of those big pizza peel looking things to make a narrow slide the will usually get wavy due to variation in trench depth.
I mowed to basically ground height between the beds, weed whacked around the beds, and put in a layer of that thick paper builders will use to protect flooring below the mulch. Some areas for cardboard instead, but we just didn't have enough cardboard to cover it all. Hopefully it will be enough to kill the grass and hopefully that results in less grass appearing in my raised beds.
I'm feeling a little called out right now. I have a similarly chicken proof raised bed area. The grass grows taller than the plants and I have to get in and mow it. One day I found some 1x2 ft tiles on a curb. I have to put those down to kind of help control the grass but it's not helping. I feel like I need to gravel the whole area because we get so much water.
This fence has been in place for at least four years, so this has been a very long time coming. I went with mulch because getting rid of rocks is really annoying should we want to change the area again in 5-10 years.