Good for you for testing and planning.

So, what you need are sources of organic nutrients. And, in the meantime, you need food. I think there are lots of plants that tolerate that low pH. Potatoes for example. Also, blueberries will love that low pH, and berries in general prefer low pH.

If you can locate a source of free manure, that's one way to go. Rock dust might also help with raising the pH. Stone countertop makers and monument makers might have stone dust for free.

The best time to add that lime is in the fall and it is best if the lime is tilled in then. You could also till in lots of organic matter at the same time, knowing it can take time to reach the optimal 5 to 8 percent. Keith Baldwin, who was a professor of soil science at North Carolina State University, wrote the linked article on improving clay soils for Taunten Press' Kitchen Garden some years back but the information (except about peat moss) is still valid.

Here is a link that might be useful: Improving clay soils

I can't till the garden in the fall because August and September are zone 7's second spring -- time to plant all the greens that will feed us through the winter. I have to form a plan for working things in as I go along pulling spend plants and preparing soil to plant the next season's crop.

If I had space, I'd have separate winter and summer gardens so I could plant one and improve soil on the other during the off season, but I haven't space enough for as much as I'd like to grow in one season even.

I wish my soil was clay. Clay is rich and fertile if you can work with any drainage issues.

What I have is nearly pure quartz sand -- gleaming white in all its sterile glory. A couple interglacials ago, this was prime, beachfront property and these sandhills are the remains of those dunes.

I may be able to source aged horse manure.

Small fruits would be nice to have, but the space requirements are considerable, the initial capital investment is beyond us, and the return on said investment takes years to show up.

With the situation clarified, any thoughts on improving the soil in this kind of as-I-go-along fashion? :-)