Its a perennial and interesting problem...
I should check that you are intending to use this with SDS, right ? And that you are wanting to do this properly, and therefore would be discounting the quick and lazy cheats style approaches, like re-meshed boole or volume builder methods ?
If so, you are thinking along the right sort of lines, but the implementation of it is unideal so far, in that you have ended up with triangles on the inside and error-state concavity problems at the corners of the outside. You can't have concave quads like this; no internal poly angles should be greater than 180 degrees.
More info on that here in case its unclear...
So those arrow shaped polys are wrong, or at least technically bad. I would be surprised if they render OK, without any tearing.
Anyway, you want to find a density of tube segmentation so that the block sits slightly inside the topology on both sides, like so.
Of course it will be different per side. Then you inset each poly group the once, and then you are setup to do the 180 degree cut thing, as demo'd here by Shepard O'Neill a few years back.
I am mad busy this week on my own modelling nightmares, otherwise I would have demo'd it for you, but hopefully this should send you n the right direction...
CBR