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In what I'm suggesting, both the block and the main part would be printed in the same orientation again, too.

The issue isn't bridging, it's that any bridged part with a hole in it is going to come out messy, just like it did in your pictures. The easy fix would be to actually add one or two solid layers to the bottom of the hole, to be "poked out" later on. There's another trick I've seen floating around somewhere that involves adding "flaps" in alternating orientations to help the bridging work out, too, but I'd need to try and find the reference material for that.