In addition to the above references, you may be interested in taking a look at this paper:

* P. Piccione and D. V. Tausk, _On the Banach differential structure for sets of maps on non-compact domains_, Nonlinear Anal. **46** (2001), no. 2, Ser. A: Theory Methods, 245–265, https://doi.org/10.1016/S0362-546X(00)00116-4

where they study how to introduce a Banach structure on sets of maps between a possibly non-compact topological space as domain and a smooth manifold as target. Of course the regularity discussed here is less than $C^\infty$, giving you a Banach structure instead of Frechet. This has obvious advantages and disadvantages; but regardless of the differences is possibly worth looking at, given that the non-compactness issues of the domain are dealt with successfully. Needless to say, once you have the desired structure on the set of maps, restricting to the particular case of sections of a bundle is straight-forward.