I realized about 80% through that this answer may have too many parts to implement, so I'm also going to leave a shorter one.
If you prioritize filling some cells over others, you can at least prune the computer's search tree. I suggest filling the center cube, followed by the six face centers, followed by the eight corners, finishing with everything else.
Here, let me number the squares of a Y pentomino:
By starting with the Y filling the center cube, the search space can be reduced by a factor of 12 by holding this piece's orientation constant. Note that only cells 2, 3, and 4 can occupy the center of the cube. If 2 or 4 is the center cell, then five face centers will remain to be filled; if it's 3, then all six will remain.
For the five or six centers, the Y occupying each center can be placed in one of 36 ways, disregarding overlap, as follows:
-The "spine" (the cubes 1245) can point in any of four directions
-Again, only 2, 3, or 4 can fill these cubes
-The full Y can lie on the face in three ways for each cube: on either Y face for all three, plus either along the spine for 2 and 4 or alongthe furthest opposing face for 3
Now for the eight corners. Only 1 and 5 can occupy these cubes, so for each of these corners we can work on partial placements for just the spines along the edges. Now, each corner is adjacent to three three-square edges, although some of the twelve edges are certainly part-occupied by Y's placed in the previous phase. The corners to be filled need to be prioritized by fewest available edges.
-If, at any point, an unfilled corner has 0 free edges, then either the last "temporary edge placement" or the entire face-center configuration needs to be advanced to the next one
-Whenever a corner piece is laid along an edge and the other corner sharing that edge is not yet filled, that other corner should be re-prioritized for having one fewer free edge
-If some unfilled corner has only one free edge, a spine is placed along that corner and edge
-And if no unfilled corner has fewer than two free edges, pick and note a "temporary edge placement" to fix if it leads to a contradiction. (I guarantee at least two corners will have no more than two edges. If the first placement is wrong, the second placement should be used but not noted as temporary.)
This gives eight partial placements and six or seven total placements. This leaves eleven or ten Y's to place within a volume of 63 or 58, respectively, which includes at least 26 cubes that are absolutely not filled regardless of how exactly the corner Y's are placed (each one still has up to four possibilities for the placement of the 3 cube), a much smaller search space.