Do you have space to just drop it? Is the lean the problem, countered by the live limb removal, then drop?
I'm guessing that there might be more to it that the pictures show, if you are soliciting advice on the tree.???
How is the hinge wood? I'd bore it to check, bar in line with the lay, strip bark and examine, "sound" the tree with ax or maul.
If it isn't punky, you could just muscle it over a little by choking a rope around the main trunk, encircling some live tissue in the process, wedge or jack the tree with a bottle jack (no vibration from pounding wedges, and steady, beware wood compression above small top of bottle jack, maybe a piece of scrap plate steel on top of jack). Deep face cut to undermine the center of gravity more than a 1/3 face would, maybe halfway, with tensioned pull line.