Grass (lawns) are an effective competition for any plant, that's why they're so succesful in mostly sunny locations. Even trees find it hard to compete with them in their younger years.
So yes, you'll need to lift the turf and quite a way from the crown (base) of the plant to be effective. Besides, lawns need generous amounts of water to stay healthy and green while lavender thrives in dry, fast draining, poorer soils baked in sun.
I'm personally not a fan of landscape fabric. Ultimately they do nothing for you because in time weeds will just grow through and they have an adverse effect on the soil biome. You're better off mulching thickly with a coarse, natural material like wood chips