The bad news is that I disagree with the first assertion: there exist non-orthogonal DWT (biorthogonal for instance).
But the good news is that this is not important here, and the answer is yes. A Laplacian pyramid is an overcomplete expansion. It can not be treated as a basis, but as a frame. And the concept akin to orthogonal bases in frames is the tight frame.
You can get details on such constructions in Framing Pyramids (2003), whose abstract is:
In 1983, Burt and Adelson introduced the Laplacian pyramid (LP) as a
multiresolution representation for images. We study the LP using the
frame theory, and this reveals that the usual reconstruction is
suboptimal. We show that the LP with orthogonal filters is a tight
frame, and thus, the optimal linear reconstruction using the dual
frame operator has a simple structure that is symmetric with the
forward transform. In more general cases, we propose an efficient
filterbank (FB) for the reconstruction of the LP using projection that
leads to a proved improvement over the usual method in the presence of
noise. Setting up the LP as an oversampled FB, we offer a complete
parameterization of all synthesis FBs that provide perfect
reconstruction for the LP. Finally, we consider the situation where
the LP scheme is iterated and derive the continuous-domain frames
associated with the LP.
And yes, the pseudo-inverse can be less simple, but you will gain more optimality in the presence of noise.
Notably, such design allowed the construction of contourlets, a multiresolution directional tight frame designed to efficiently approximate images made of smooth regions separated by smooth boundaries.