Vertical 5 mm diameter hole comes out fine, but diagonal 3 mm holes keep collapsing
A member of the public on Stack Exchange's forum gave me the answer:
&quot;The problem is not the hole 'collapsing' in and of itself. The problem is the material.
Versatile Plastic is a layered powder fused together with a laser in multiple layers, so the unused power has to come out from the various holes in the model once printing is completed. However, these layers are also very thin and compact. Therefore, if you make a small enough hole, the powder simply won't flow through it and it becomes stuck. Due to the small dimensions you listed (of said hole) it's gonna be difficult to free it due to the sheer compressing forces involved. When working with such fine details Versatile Plastic is not the most indicated material.
The only correction you can make to rectify the problem is changing material. Fine Detail Plastic would be a good choice for a new test, although it is not cheap compared to Versatile, is certainly cheaper than metals.
Fine Detail Plastic should provide you with a good result if you feel like trying it out.&quot;
ShapeWays technical support told me basically the same thing, and added that ordering the shape in black might also exacerbate the problem:
He said that when a model has holes in it, a human actually pokes the holes to get residual material out if possible.  Then he adds: &quot;The last issue would be that Black indeed causes additional issues. If the production team isn't able to clean the model properly, or didn't do their work right, this could become and issue when we dye your model. All Versatile Plastic is printed in White, when selecting a colored finish such as black, we dye the model using a bath of regular RIT DYE. If the hole still has trapped powder, the bath will make the loose powder wet and make it become sticky, in combination with the dye this causes it to become chunky on the inside.&quot;
TL;DR: I can try to fix it by using Fine Detail Plastic, or I could try ordering in white and then painting it black myself to avoid gunking up the holes (the insides of the holes don't need to be painted black).