While biological structures are ultimately 'DNA-controlled', the translation of the ACTGs to codons to proteins to structures explodes in complexity. Even within that simple molecular (i.e. no microstructures, let alone macrostructures like the whole cell) realm, everything is influencing everything else all at once. Proteins will suppress or enhance transcription of others, alter how other proteins are synthesized, and so on.
Once you have some molecules/proteins, they'll assemble into microstructures controlled by other proteins and small molecules that can push/pull on 'things', and they'll be influenced by the ambient conditions: temperatures, pHs, mineral contents which they're not fully in control of, but they're the progeny of ancestors that have been doing it for millions of years in the same habitat, and they've been trained and optimized to handle those varying conditions, so they get it right most of the time.
The biology then interacts with physics and chemistry to kinetically grow faster here to make something grow outwards, slower here to make it grow inward, then you can start forming shapes.
Diatoms reproduce by separating the two halves of their shell and regrowing a new half. It seems to me that it would be difficult to predict the shape without an existing half shell. If anything, the shape is probably mostly determined by the shape of the existing shell; if a parent is damaged and a chunk is missing, you want to make a new half shell that seals up against that half instead of the shape it's "supposed" to be.
I think you're right. The DNA does control the shell shape, but the way DNA works is always thru chain reactions (secondary effects) rather than something akin to a blueprint of any kind. So if you took out the DNA and put it in the "wrong half-shell" (using your concept), I bet it would be unable to complete the other half shell that looked symmetrical, and I bet it would die, because without the symmetry it cannot "complete" the shell, and the outside environment would therefore seep in, corrupting the cell internals, and it would die from that contamination.
You could create a company called Diatomics and sell them (the 3D prints). Would make great wall decorations. Would be cool to try to use AI to try to write a program to generate the 3D datasets. OpenAI-01 is so smart I bet it could create some diatom-modeling 3D code.