I had a look at the 2x3 rectangle and the P-hexomino.
Drilling only one hole through the 2x3 rectangle we get 5 distinct pieced with a total volume of 30,
but they don't fit into a box as a exhaustive search shows. There are 20 pieces with two holes and a 4x3x10 box can be made.
Can you find constructions for other boxes or prove them to be impossible?
The 12 P-hexominoes with only one hole can fill the three boxes 2x6x6, 3x4x6 and 2x4x9 and the two rectangles 1x6x12 and 1x4x18.
The 2x3x8, 2x3x12 and 2x2x18 boxes are impossible, because there is an hole of length 4 that must be parallel to the longest edge.
But only one hole of length 2 can be used for extensions. Therefore a construction fails.