By a topological (2,2)-flip Dougherty, Faber, and Murphy mean a bistellar move on 2-manifolds (and not on 3-manifolds). So it looks like the example
on a 3-simplex, a (2,2) non-topological flip would map one edge of the tetrahedron into the edge it does not intersect.
is supposed to be about trying to apply a bistellar move to the boundary of a 3-simplex. It further looks like the geometers want to see the result of such a generalized bistellar flip as the 2-disk, triangulated by the suspension over the 1-simplex. As a topologist, I'm deeply troubled by such vision of a 2-dimensional Pachner move (e.g. what of a "flip" does it retain if so generalized?) I would either think of this as a 3-dimensional move indeed (see more about this below) or else I would consider the following to be the result of this move: two copies of the suspension over a 1-simplex, glued along their boundaries. This is no longer a simplicial complex, but a "pseudo-complex" in the sense of the Hilton-Wylie textbook, and a "singular triangulation" of a more modern tradition.
Certain generalizations of bistellar moves to singular triangulations have been studied by Matveev and his students; they are noteworthy at least for being precisely dual to Matveev's moves on special spines (concerning the duality, see for instance this review, though there must be better references). Some of these generalized bistellar moves are not supported by homeomorphisms (so maybe geometers would call them "non-topological"). For instance, they have a move that collapses the join of a 1-simplex and S, where the 1-sphere S is the union of two copies of a 1-simplex along their boundaries, onto the suspension over a 1-simplex. This 3-dimensional singular flip can be decomposed into a sequence of two "(2,2) non-topological flips".