Ensure groups of four 3-tuples have 9 unique numbers
I suspect finding the optimal assignment will be NP-hard in theory and possibly hard in practice, but it might be possible to find a near-optimal solution.

One approach is to express this as an instance of integer linear programming, and use an off-the-shelf ILP solver.  Let $x_{i,j}$ be a zero-or-one integer variable, with the intended meaning that $x_{i,j}=1$ implies the $i$th tuple is assigned to the $j$th bin, and $y_{j,d}$ another zero-or-one variable, with the intended meaning that $y_{j,d}=1$ means that digit $d$ is covered in bin $j$.  Then you can write down a set of linear inequalities on the $x$'s and $y$'s that captures what constitutes a consistent and valid solution.

For instance, the constraints might be: $\sum_j x_{i,j} = 1$ (each tuple is assigned to one bin), $\sum_i x_{i,j}=4$ (each bin receives 4 tuples), $y_{j,d} \ge x_{i,j}$ if the $i$th tuple contains digit $d$, and $y_{j,d} \le \sum_i x_{i,j}$ where the sum is over all tuples $i$ that contain digit $d$.  Now you want to maximize the linear objective function $\sum_{j,d} y_{j,d}$.  Then, you could use an off-the-shelf ILP solver to try to find an optimal solution to this linear program, or ask it to report the best solution it can find within a given time bound.

I don't know if there is a better approach.  You could look at algorithms for exact cover to see if any of them can be adapted to your setting.