Fine lets use the Cayman Islands as an example then?
Net importer of food, total imports of energy, 100% depended on fossil fuel.
You can definitely get mining going with solar panels, a tidal array, wind power, and even biogas would be on the table
Can you get energy for those at 2.5 cents per kWh, that's the price coal and gas flaring gets you?
Of course, you can get it with government subsidies, but in this case, the government will be you, so more money to throw away.
I mean the dutch turned the flevopolder into arable farmland and this is nowhere on the scale of that.
With the difference that most of those that are suited for agriculture are the podlers and fenlands along some marshes, much of the land used to compact the drained soil was rich land from the estuaries, of which there are a ton, on your land, you won't have one!
I'll not cover the other part about soil since you're talking about home gardening I'm talking about feeding thousands, once you go out of your garden and try to do the same on at least 1ha of land you're going to see the limits of home-brewed compost and no pesticides

Possibly, with the state of the art process using cofferdams and vibro compaction, you could do it for maybe around a billion dollars. The price tag sounds pretty steep but its not really all that bad in regards to material moved nor price.
Seeing how 45m mc3 project costs 1.2 billion and your island would need that only for one foot depth (or height) at 49 square miles I think you've missed a zero