You want the flat edge to just rest on the neck,/frets, not touching the nut at all. You may now have taken too much height off the bridge, though if the string tension lifts the top up, then things may balance out.

9mm is probably far too thick if you are sticking the block under the bridge on the inside of the guitar. Think about how the strings are actually held in place by the bridge pins. The bridge pin pushes the ball end forwards, so that it is half-resting on the underside of the top. It's this that keeps the string in place, not friction from the bridge pin.

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So you don't want the depth from the top of the saddle to the bottom of any bridge plate underneath the saddle to be too deep, otherwise the bridge pin won't be long enough to push the ball end forward and the string will keep pulling out. You ideally want that overall depth to be about the length of the twisted section of the string above the ball-end, so roughly 15mm. A bit deeper is OK, but probably no more than 20mm as you want the bridge pin to extend beyond the ball end.

You don't want it too shallow either, as you don't want the thick twisted end of the string to extend above the top of the bridge as it won't bend easily and so (especially on the wound strings) you'll end up with a much shallower break angle over the saddle. So 14-15mm is probably the shallowest overall depth you want.

You should be able to stick something through the hole and mark up on each end where it emerges to see what depth you currently have, and work out what the thickest base plate you can use is.

You can certainly make your own Bridge Doctor style support, but note that you really do need the ability to adjust the length under stress of the post running to the rear block, rather than having a fixed length that just rest against the back (which may also cause buzzing if the top lifts and pulls that post away from the block slightly).

It's the ability to screw in the thread that presses on the rear post that makes the Bridge Doctor work, so anything you do needs to have that ability, and you need to be able to adjust that from the sound hole.