Ok, it seems there is some confusion, because I haven’t explained my setup well.
This (simplified) picture shows how things will look after the 1st setup is done. Note the two faces that were milled, the top was also faced off. This means that the figurine inside the box is fixed with regards to the corner reference point. The exact stock dimensions (or stock squareness) do not matter.
Now, this point will become our WCS zero after flipping the part, this picture shows it without actually flipping the part, perhaps this makes it easier to understand.
So, in the second setup, I’m looking at this:
I am not re-zeroing the machine between setups (although that again wouldn’t matter that much here). The important thing is that for the 2nd setup, the milled corner needs to fit tightly into my blocks, and most importantly, my machine needs to be zeroed properly, so that WCS origin is exactly where the blocks and the threaded table meet and where the milled corner from 1st setup will land after flipping.
I don’t think milling 4 sides would change anything here. I am milling three faces that meet, so I do have a good reference corner. Those three faces should be perfectly square after the 1st setup and all other geometry should be correct with respect to those faces.
The issues are on X and Y, to various degrees.
I still think BitZero is the problem. I just did a quick check: zeroed in on the corner of that woodworkers square, then jogged the pin close to the square — it was visibly off, this time more in X than Y.
I also tried a manual approach: I used the BitZero as a manual contact indicator by keeping it pressed against the tiger clamps (and blocks) and jogging the Nomad until contact is made and the LED goes red, then zeroing X or Y and offsetting by 1.5875mm (half the diameter of the dowel pin). This got me much better results: at least visually, the blocks were smack dead in the middle of the pin.
That electronic edge finder looks good, @Moded1952 — do you use that with the Nomad? I can’t see the shank diameter in the specs. Price might be acceptable, but I’m looking for something that I could use right away (like I hoped the BitZero would work), ideally as automated as possible. In other words, I’d rather not hack on G-code probing routines 
There is also another interesting probe here: Touch trigger probe TPA2
Unless I get better suggestions, I think I will try running some jobs using this manual method I just used and see where that gets me.