Amos Altshuler studied a related notion in his Ph. D. thesis and the paper "Altshuler, Amos Manifolds in stacked $4$-polytopes. J. Combinatorial Theory Ser. A 10 1971 198--239." In his version he allows manifolds and not only spheres and demand that the manifold contains every edge. (He consider simplicial 4-polytopes but thos can be realized as simplicial complexes in 3-space.)
I think I saw some subsequent papers on similar notions of "hamiltonian manifolds and spheres" but I could not track them. (I think this is Schulz ' paper)
Other related papers (still not those I thought about) Effenberger, Felix; Kühnel, Wolfgang, Hamiltonian submanifolds of regular polytopes. Discrete Comput. Geom. 43 (2010), 242--262.
Schulz, C.: Polyhedral manifolds on polytopes, Proc. Conf. Palermo 1993. Rend. Circ. Mat. Palermo (2) Suppl. 35, 291--298 (1993)
The Mathscinet review by David Walkup reads:
M is said to be a “good” manifold in L if L is a simplicial complex, M is a subcomplex of L
containing all the edges of L, and |M| is a closed 2-manifold. The case when |L| is a 3-sphere,
especially if L is the boundary complex of a simplicial 4-polytope, is of interest. A “stack” is
defined inductively as follows: A 4-simplex with its faces is a stack, and the union of two stacks
is a stack if their intersection is the closure of a common 3-simplex. If K is a stack, then |K| is a
4-ball, the boundary complex Bd(K) is well-defined, and Bd(K) can be realized as the boundary
complex of a simplicial 4-polytope. Any good manifold in a stack K must lie entirely in Bd(K).
A “star” is a stack obtained by stacking a 4-simplex on each of the five 3-faces of a central 4-
simplex. Theorem 7: If K is a star, then there are exactly 6 good manifolds in Bd(K), each is
a torus, and all 6 imbeddings are isomorphic. Theorem 14: For any n >= 1 there is a stack K
of 6n 4-simplices which can be obtained by stacking together n stars so that Bd(K) contains a
good manifold of genus n. Conversely, Theorems 16 and 4: If M is a good manifold of genus
n in a stack K, then K must be the result of stacking together exactly n stars. Other theorems
characterize those ways of stacking stars so that the result admits a good manifold. Theorem 23:
The maximum number of different good manifolds in K as K ranges over all stacks with 6n 4-
simplices is 6, 8, 12, 24, 40, 80, or 2n according as n = 1, 2, 3, 4, 5, 6, or n >= 7. Theorem 26: For
every g >= 1 there exists a 3-sphere which cannot be realized as the boundary of a stack but does
contain a good manifold of genus g. Many other nice results and fruitful ideas are developed.
The MR for Schulz paper: The author gives a survey on polyhedral manifolds which are contained in the boundary complex of convex polytopes. Manifolds with certain extremal properties (e.g. minimal number of vertices) are of particular interest. Another fruitful field is Hamiltonian $2$-manifolds (i.e. $2$-manifolds containing the $1$-skeleton of the polytope). The author gives a wealth of material by several authors from Möbius, Coxeter, Ringel, Jungerman to Altshuler, Brehm, Bokowski, Kühnel, McMullen, Schulz and others.