The short answer to the title question is likely yes. There are two separate issues discussed in the OP, one is more philosophical on Gauss's disagreements with Kant's apriorism about space, and the other, concerning his insights into the orientation of surfaces and non-orientable surfaces. The first question is addressed to some extent in Was Kant a factor in forming Gauss's abstract view of mathematical objects?, so I will focus on the second.
Gauss corresponded with Gerling for a long time, at least since 1819, when he mentioned the angle sum defect ("Gauss-Bonnet theorem") for hyperbolic triangles in a letter to him. At the time, the main subject was non-Euclidean geometry, and this is discussed at length in What did Gauss read in the Appendix? by Abardia, Reventós and Rodríguez (the reference is to Bolyai's Appendix from 1832). Issues related to what Gauss called theorema egregium (invariance of curvature under isometries) are discussed in 1825 letters to Schumacher. Gauss's celebrated work on geometry of curved surfaces Disquisitiones Generales circa Superficies Curvas came out in 1827. His relevant writings on the subject, both published and unpublished, are discussed at length in 150 years after Gauss’ «Disquisitiones generales circa superficies curvas» by Dombrowski. Dombrowski quotes an unpublished 1825 draft of Disquisitiones Generales, which shows that Gauss possessed the concept of surface orientation in 1825 (and possibly as early as 1794). In his modernized "free translation":
"The sum of the angles of a (small) geodesic triangle Δ in a curved surface in $\mathbb{E}^3$ is equal to the sum of π and the oriented surface area of the spherical image of Δ, where the oriented area is taken to be positive or negative according to whether the boundary of the spherical image of Δ winds around the image in the same direction or in the opposite direction as the boundary of Δ winds around Δ." (57)
Gauss then generalizes it to geodesic polygons. Dombrowski discusses the context and speculates that Gauss was led to the general statement for curved surfaces by the hyperbolic case combined with his practical experience with geodesic measurements in 1812-1822:
"The result (57) was already widely known in the special case of a developable or a spherical surface. Gauss possessed the result analogous to
(57) for hyperbolic geometry (and therefore essentially for surfaces of constant
negative curvature, the author) already in 1794 (see G.W. 8, p. 266), and announced it to Gerling in a letter in 1819 (see G.W. 8, p. 182). It is therefore easy to imagine that Gauss was led to a geometrically intuitive "insight" concerning the validity of (57) for the general case of a surface of non-constant Gaussian curvature from the knowledge of this fundamental case and on the basis of his rich differential-geometric experience in geodesy (with geodesics and trigonometry spheroids, but also with questions of mapping and bending) obtained in the years
1812 to 1822.
[...] Although Gauss had studied intensively the concept of "oriented surface area" as used by him in (57) (see G.W. 8, p. 398, line 2 from below), he did
not think that his corresponding studies had "matured" sufficiently... The proof (with its so impressive geometric arguments, which we have just retraced from the fragment (55)) was never published by Gauss! The reason for this lies on the one hand certainly in his self-criticism of his own above mentioned sketch of a proof for (57), and particularly of his concept of the "oriented surface area of the spherical image on a curved surface" (involved in that proof) which he has not defined with that analytic rigour usually applied by him."
This result leads to theorema egregium discussed in 1825 letters to Schumacher. The Gauss-Bonnet theorem itself is brought up in the 10 October 1846 letter to Gerling (Dombrowski's translation):
"The theorem which Mr. Schweikart mentioned to vou,that in any eeometrv the sum of all outer angles of a polygon differs from 360 by a quantity,..., which is proportional to the surface area, is the first theorem lying almost on the threshold of that theory, a theorem whose necessity I already recognized in 1794".
So the appearance of orientation related issues in the letters to Gerling and Schumacher in 1846 (connected to the older themes of his agreements and disagreements with Kant) is not surprising.
As for the non-orientable surfaces, the issue is much more murky. The "Möbius band" is described in private papers of Listing and Möbius in 1858, almost simultaneously and in similar terms, Listing's description is earlier by several months. Gauss died in 1855, but both were Gauss's students, and many historians attribute many of Listing's topological ideas to Gauss. Listing himself stated something to this effect, although without specifics. Biggs in his chapter in Möbius and his band explicitly speculates that Gauss might have been the common source here, but admits that the issue is undecidable:
"Both of them describe the construction in
very similar terms. Is this another one of those istances which sometimes
happen in scientific discovery, where an idea whose time is ripe
appears iI1dependentiy in different places but at the same time? That
is certainly a possibility. Or was there a common reason for the fact that both Möbius and Listing described the Möbius band around
the same time? If the latter, then the likelihood is that the common
reason was connected with the work or Gauss who, as we know, had
been very interested in this kind or topic. Gauss had died in 1855, so
the idea cannot have been communicated by him directly, but the
possibility of some link with his work remains. I doubt that the question will ever be completely resolved."
