I saw this thread late, and you've probably already made you decision. But here's what I typed up before I got that far in the thread:
First, there's the principle of "if you can't extend the root zone DOWN, build it up. That's what the containers sitting on top did, gave you free-draining root zone ABOVE the clay.
If you accept that the bed will never have well-draining soil much deeper than the grade of the soil outside the bed, you HAVE to build it up or grow only shallow-rooted plants. Maybe use boards wedged between bricks and soil to imitate taller walls and support more soil.
This next idea would be expensive or a LOT of work, so not practical: If someone offers free bricks, add a few rows of brick on top (with or without mortar). If you have rocks that can be stacked neatly to make the walls a little taller, there you go.
It sounds like draining STRAIGHT DOWN will never work. How about draining "mostly sideways"? Like an almost-horizontal French Drain that runs at a shallow angle from the bottom of your bed to a lower spot ... or at least a long enough trench that, after it fills up, water CAN perk down out of it fast enough to provide a basin for your bed to empty into.
(One bed I have drains into a nearby hole that's large enough to hold most of one rainstorm's worth of water from that whole bed. The hole usually drains mostly empty between rainstorms during the growing season. Your "horizontal French Drain" would be like my drainage hole - a holding zone for runoff, a little deeper than the bottom of the root zone you're creating.)
However much you dig the bed down INSIDE the brick wall and amend it, you should probably also dig down and amend the soil surrounding the OUTside of the brick wall to the same or slightly deeper depth.
Then, assuming the brick has some cracks, or you drill some holes, or the amended soil extends lower than the bottom of the bricks, water inside the bed will wick through and under the brick wall. In effect, it would drain "sideways" into the soil outside the bed.
Now you have to make sure that the soil outside the bed has some channel or trench to carry excess water away from the bed. That channel might need to be mostly below grade, hence I suggest just digging amending a trench rather than your whole backyard.
The hard clay "floor" of the bed should slope down a little as it gets farther from the house, to encourage drainage away form the house.
The amended soil on the outside of the brick wall can be narrow, if it connects to a trench that continues the downward trend away form the house. I make these mini-drainage trenches barely wider than my mattock blade, and that seems enough to handle a medium amount of rain. Back fill the trench with very amended soil that drains adequately (or some people would fill it with drainage garvel".
The bricks may stay more stable if you don’t undermine the entire wall with amended soil. I would amend everything inside the bed, plus a strip outside the bed, but only undermine around 30-50% of the wall. leave 50-70% of that hard, hard clay in place to support the wall. Just tunnel under a few places in the wall (short trenches) to let water out of the bed. A shovel's-width ought to be enough for each "mini=tunnel".
Amending those tiny tunnels with organic matter that wicks is extra important. Bark, compost, grit and coarse sand are what I would use. Coir if you have it.
