You are throwing out topsoil and organic matter with the lawn Nick. Not a great plan.
There are a few things you could do. The easiest being to spray the area with glyphosate and waiting a month or so for the grass to die then just digging over the area, do the soil improvements you need to and
plant away.
Alternatively, and my preference, mow the lawn very short, sprinkle it with pelletised chicken manure and bury the lot (after a good watering) with 10 sheets thick of newspaper (or cardboard).Don't leave any gaps. Then mulch the lot. Heavily. Water again well. The grass will die in a few weeks because of lack of light. It will add organic matter to the soil and critters will break down the paper and do your digging for you. I've made all my gardens this way, a few needed further additions of organic soil and compost as well as some lime but the effort was minimal generally.
This is a basic idea of a widely used concept and there is heaps of info available on "no dig" gardens. Once you've got the concept understood, you can adapt it to your situation.
Oh BTW. Don't use carpet to kill the grass.