Cut the grass out of the soil and put to one side. Cut it in little squares the size if your spade. Then dig the trench with a pick or maddock and trenching shovel. A crowbar may be handy if the soil is hard. Take care of your back. Once you come to a root, sever it each side of the trench with a very sharp spade ot the crowbar. Continue until you are a few inches deeper than the deepest roots. "Stand " some rootgaurd in the trench. Sort of have it too high so it hangs out the top of the trench and you can weigh it down with soil. Then backfill the trench, cutoff the excess rootgaurd and replace the grass turf. You'll never know you were there. The roots that are in the garden bed will rot after being severed unless they are a type of
tree that sends up 'suckers' in which case they will need to be removed.
Shadecloth or weedmat would possibly suffice if you can't get rootgaurd.