I think you and I are on the same track. I am growing garlic for the second year, and I am also hoping to incrase my crop every year until I have several hundred plants. because I've been using seaweed as mulch it has done really really well. I live in Nova Scotia, so we have pretty cold winters. But I am also leaning towards living mulch myself this year. The thing about garlic is that it is a pretty heavy feeder. It needs a lot of nutrients, and those seem to be well supplied by seaweed, although I'm sure there are plenty of better-informed people here who could tell you exactly what it needs. What I was planning on doing differently this year was scattering the garlic greens back on the bed sometime after harvesting. The only issue I see with that is that garlic keeps longer if you don't cut the greens off, so I will want to keep them on as long as possible. So I'm not sure how I'm going to work around this.
In terms of a cover crop, you might also consider buckwheat. When this dies off or gets cut it makes GREAT
straw, which is one benefit you do not get from clover. It does make sense not to plant cover crops that are going to compete, but if you stagger planting so that you sow the cover only after the garlic is well established then maybe it will work out. Paul Wheaton posted a
video very recently of his friend in Missoula who uses clover as a cover crop and she had the same concern, but she does mention how they dealt with this, so you might want to watch that video too.