Let's say we have a shape. We will call this shape $A$ and we will say that $A$ is some finite subset of the regular square tiling. Similar to a polyomino, except we do not require that $A$ has a connected interior.
Now I would like to a the optimal way to pack copies of $A$ on the plane without rotation or reflection. For example if $A$ were:
one optimal packing would be:
another would be:
despite the fact that if we were allowed to rotate or reflect $A$ we would be able to tile the plane perfectly.
I've been trying to come up with an algorithm to find an optimal packing for arbitrary $A$, but I am rather stumped and looking through literature I was not able to find any work on this problem.
The disconnected polyominoes you refer to are more commonly known as animals on the square lattice, though some authors frustratingly use this terminology to refer only to polyominoes. "Translational packing density" is the term I'd use for what you are looking for here, though I agree that it seems there is not much research in this direction.
The problem is likely fairly difficult, or at least not one for which provably correct algorithms are known. Not until Bhattacharya 2016 did we even have a provably correct algorithm for the decision problem of whether this density is equal to $1$ (i.e. whether the animal tiles the plane), and that only via the fact that such tilings must be periodic, which does not lend itself to nice runtime bounds - we just alternately check for periodic tilings of a certain size and try to exhaustively prove impossibility within a certain radius, until one of the two avenues succeeds.
The more general optimization problem in this question seems unlikely to have been solved. Of course it is hard to guarantee such things, but given the recency of more specialized results than this and my not having seen such results in a reasonably wide survey of polyomino-adjacent literature, I would be surprised to learn that any effective algorithms were known.
(Although empirically, it doesn't seem to be too hard to get optimal results quickly just by trying small periodic tilings - in the case of polyominoes, I verified a while ago that the optimal translational packing densities all result from isohedral arrangement of the polyominoes (i.e. with their centroids in a lattice) for every polyomino with at most $6$ cells.)