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Adding dot2tex to the Sage installation, as suggested by FrédéricC and John Palmieri, did work for me and produces Hasse diagrams with correct layout, and non-verbose LaTeX.

One should

1. Install graphviz to the OS (using the appropriate package manager)
2. Install dot2tex to the Sage directory using ./sage --pip install dot2tex

This assumes that Sage is compiled from source; I don't know how one would add dot2tex to a distribution package of Sage.

Thank you FrédéricC and John Palmieri!