IMO, a groundcover is not a great idea for a perennial bed. The whole point of a GC is to spread rather aggressively to cover and you will often encounter the problems you have experienced - most are too aggressive and overwhelm the perennials, especially those that are fully herbaceous or die to the ground in winter. Sweet woodruff hitchhiked into my woodland garden with a gifted plant and I spent years trying to eradicate it, so I understand your frustration :-) An incredible pest!!

I think just mulching the bed is a better option, especially as many perennials will gradually expand over time to fill in any gaps. Or you can try some low, spreading, groundcover-like but well-behaved perennials to fill in, like some of the hardy geraniums or Veronica 'Georgia Blue'. I like deadnettle, Lamium maculatum, in a woodland planting but some find it too spreads too aggressively for them - I've never had that issue with it. One of the creeping wood phlox, a saxifrage or native hardy gingers could work also.

It sounds like what you have is galium triflorum (fragrant bedstraw), not galium odorata (sweet woodruff). I use sweet woodruff as "green mulch" it spreads with *enthusiasm* but is very shallow rooted and easy to remove/control, and does not choke out my perennials. Much better behaved than most groundcovers.

I'd check for you climate and zone, but I'm finding Irish Moss (it's neither) very polite, and I'm trying out "brass buttons" and ornamental strawberry as ground covers.