I started a garden a little bit smaller than that size last year. I did hire a rototiller guy to work up one area that was for finer seeds. It was very successful, despite a drought in the spring/early summer and too much rain in the later summer. I hand dug some areas for finer seeds, and to be honest, it was a lot of work for the results though it was lovely and almost weed free.
If I were you, I'd get the cardboard on as soon as possible. In the snow, even. I put my cardboard on for a new section I am starting for this spring in the fall. I anticipate it will have broken down and softened a lot by spring, and will keep any grass from sprouting. I also wandered around the community picking up bags of leaves. Cardboard under mulch (mostly
yard waste I picked up, including grass trimmings) pretty much dissolved to bare earth by fall. Note that most herbicides for residential use are illegal where I live, other places may need to be more careful.
With minimal compost, I would reserve it for rows for starting things like carrots, etc, that are finely seeded. I might also keep an eye out for ads for manure. I bought manure and mixed it into my rototiller area, and top dressed around my corn and squash. Garlic would also like better soil. Some people have manure free if you shovel and load yourself, they were all 1 hr+ drive from me, so not worth the gas for the amount I can fit in my car to me. I topdressed homemade compost or compost tea around other things like tomatoes and peppers. I also bought straw as additional mulch which was great in the drought and pretty much disappeared by fall. The soil in the garden seemed richer, better textured, and more alive by fall than it was in spring.
Here are my threads about it...
https://permies.com/t/137014/Tips-year-successful-garden-beds
Also:
https://permies.com/t/148067/Successes-year-garden