Do pseudovectors also transform differently to vectors under spatial dilation, not just reflections and parity?

Well, one possibility is that this is just a matter of where attention is focused. Since, excluding the weak force, all physical laws are invariant under parity, it's interesting to note that pseudovectors transform differently than vectors under that operation. However, real physics isn't, as far as we know, invariant under spatial dilations, and so for most people this isn't an interesting (approximate) symmetry and likely not included in the mental set of operations one could apply to vectors or pseudovectors. Perhaps researchers of conformal field theory have a different point of view.

A second point is that $vec{L} = vec{r} times vec{p}$ is just one equation for one pseudovector. By dimentional anslysis, the angular momenentum is proportional to the length scale squared, and so this double-dilation could be viewed as just its dimensional properties. Pseudovectors are, more technically, elements of the exterior algebra of a vector space, whose basis elements are the wedge products of basis elements of the regular vector space. Since physical space has dimensions, so do its basis elements, and thus the basis elements of the exterior algebra are dimensionfull as well.