Drainage is not about plants it is about bed preperation and so dig down quite far doa layer of gravel then your soil for planting that way there will be enogh palce for the water to go . You will need to dig down quite far so be prepared for that job . It also would not hurt to plant things that like wet feet

Why do you think the bed needs edging? Does the soil wash out when it rains? A crisp cut slit edge is plenty. Artificial edging just makes mowing more complicated. If you plant the bed more densely the surface will better protected and less likely to be affected by rain.

You do not need to dig out the bed and add gravel unless the soil is remaining waterlogged.

@floral_uk z.8/9 SW UK -- I tend to lose some mulch in heavier rains. It's also a little tricky to trim around the flower bed without shooting mulch everywhere with my trimmer.

Have you tried edging shears? They won’t disturb the mulch and if the slit edge is deep enough and sharp enough there shouldn’t be a problem. And, as mentioned above denser planting would keep the mulch in place.

I would remove that strip of grass and add a groundcover that doesn't need mowing.

If you can get pine straw, it doesn't wash out like mulch.

Your question is not completely clear. You want recommendations on an EDGING product to apply to the sidewalk-side of the existing mulched bed?

It would make more sense to do away with the grass (make it all part of mulched bed) and apply a very short retaining wall of landscape blocks adjacent to the sidewalk. You’d have to backfill in a bit to make the bed one level. Like this.