We talking broadfork or run of the mill garden fork? I've tried the later, and well, it's more work than I wanted with the limited time I have. Area to cover is going to be a huge factor. Don't get me wrong, the results of the fork were soooo much better than tilling, but switched to an initial till followed by hand work to get the areas I really wanted improved to my specs.
I just got done putting a garlic bed in a few hours ago actually. Where I put mine was terrible clay (backfill from a recent septic system overhaul) so I went with a raised bed using cinder blocks 1 high. I cut the sod out, forked the clay soil, added compost, then topped it off with a commercial garden soil. I figure it should help with the actual harvesting of the garlic next year. To be honest if that portion is harder than it should be I'll never do it lol.
I'm leery of biochar, but I have the opposite problem you have, terribly alkaline soil so you should be in the clear. All in all sounds like a great plan.
Is your tiller a walk-behind? I find them suitable for land previously worked up, but if you're breaking fresh ground with them they suck lol, I spent more time being dragged around by it and fighting it than tilling soil. I had my dad bring in one on the tractor and the results were a night and day difference to say the least.