My understanding is that it's not the char that's alkaline but the
ash that comes with it. When I make it I don't worry at all about the ash because it's really acidic where I live. It's kind of a 2 for 1 deal. But, if you were purchasing the biochar I would think they would have much better quality control so I would think it would be less alkaline?
There are totally amendments you can put in to acidify your soil.
Also, unless it's airless, solid clay I might not worry too much about digging anything in. A thick layer of
wood chips on top of the soil does amazing things...
Also, also, horse manure is my go to for organifying clay to plant a tree in it.