I do not know how, but I missed the function
nullspace
from the LinearAlgebra standard library. Maybe it is not exactly what I was asking for, but I do not think I would have ever asked the question if I had found it before. I will thus mark this answer as a solution.
There is also
nullspace_right_rational
from the Nemo.jl package.
Anyway, maybe it is cleaner for the forum to delete the whole topic (I do not have the permission to do so…)