Geometric Interpretation of BRST Symmetry
Comments to OP's question (v1):
In superfield formalism, there is a long tradition in the literature to consider constructions that interpret geometrically BRST (& anti-BRST) transformations as translations of Grassmann-odd $\theta$ and $\bar{\theta}$ coordinates in various physical systems, see e.g. Ref. 3 and references therein. The earliest articles seem to be Refs. 1 & 2. (We caution that BRST supersymmetry should not be conflated with Poincare supersymmetry.)
If we are not allowed to introduce Grassmann-odd $\theta$ and $\bar{\theta}$ coordinates, then it seems that OP's quest for a "geometric interpretation" becomes just a matter of providing explicit, manifest coordinate-independent, differential-geometric bundle constructions for the BRST formulation of various gauge theories. This will depend on the gauge theory. E.g. Yang-Mills theory, BF-theory, string theory, etc.
References:
S. Ferrara, O. Piquet & M. Schweda, Nucl. Phys. B119 (1977) 493.
K. Fujikawa, Prog. Theor. Phys. 59 (1978) 2045.
C.M. Hull. B. Spence. & J.L. Vázquez-Bello, Nucl. Phys. B348 (1991) 108.
Construct a principal bundle with base M and structure group G. Define Projection map and trivialization of the bundle in usual way. Denote this bundle as $P_1$. Construct another trivial principle fiber bundle $P_2= P_1\times G$ with base $P_1$ and Structure group G with trivialization of $P_2$ consisting of $P_1$ and an identity map from $P_2$ to $P_1\times G$. Construct $P_3$ as $P_2\times G$ as in the second step with local trivialization consisting of $P_2$ and an identity map from $P_3$ to $P_2\times G$.
BRS transformations are identified with infinitesimal gauge transformation on $P_3$ with parameters related to ghost fields, where these ghost fields are identified with part of certain one-forms on base space $P_2$. For details consult ref. 1 and 2.
There is another approach of group manifold in which you can gauge the algebra of $G+Q$ to obtain BRS transformation of Gauge fields where $G+Q$ has the structure of a group manifold. In short, BRST transformation are a sort of diffeomorphic invariance of this group manifold. Consult ref. 3 for details.
1- Geometric structure of Faddeev- Popov fields and invariance properties of gauge theories: Quiros, Urries, Hoyos, Mazon and Rodriguez.
2- Geometrical gauge theory of ghost and Goldstone fields and of ghost symmetries: Ne'eman and Thierry-Miec.
3- Supergravity and superstrings (a geometric perspetive): Castellani, Auria, Fre (3 vol. set with first vol. containing the necessary Group manifold machinery).