There is indeed a pseudodifferential calculus that can handle this situation. It is called SG calculus, see e.g. here, or scattering calculus, see e.g. here. Symbol estimates (on $\mathbb R^n$) are
$$
|\partial_x^\alpha \partial_\xi^\beta a(x,\xi)| \leq C_{\alpha\beta}
(1+|x|)^{p-|\alpha|} (1+|\xi|)^{m-|\beta|}.
$$
This calculus can be introduced on manifolds $\boldsymbol X$ with cylindrical ends (also called SG or scattering manifolds). Pseudodifferential operators act between weighted function spaces. The Fredholm property of an operator $A$ is equivalent to its ellipticity, where the latter means (for a classical operator $A$) pointwise invertibility of (each component of) the triplet
$$
(\sigma_\psi^m(A),\sigma_e^p(A),\sigma_{e,\psi}^{p,m}(A))
$$
(i.e., it is in fact ellipticity in a calculus with symbolic structure). In case $X=\mathbb R^n$ and $A=a(x,D)$,
$\sigma_\psi^m(A)(x,\xi) = \lim_{\lambda\to\infty} \lambda^{-m} a(x,\lambda \xi)\,$ for $(x,\xi)\in \mathbb R^n\times(\mathbb R^n\setminus0)$ is the usual principal symbol,
$\sigma_e^p(A)(x,\xi) = \lim_{\mu\to\infty}\mu^{-p} a(\mu x,\xi)\,$ for $(x,\xi)\in (\mathbb R^n\setminus0)\times\mathbb R^n$ is the principal "exit" symbol,
$\sigma_{e,\psi}^{p,m}(A)$ arises as a compatibility condition between the former two:
$$
\sigma_{e,\psi}^{p,m}(A)(x,\xi) = \lim_{\mu\to\infty}
\mu^{-p} \sigma_\psi^m(A)(\mu x,\xi) = \lim_{\lambda\to\infty}
\lambda^{-m} \sigma_e^p(A)(x,\lambda \xi),
$$
$(x,\xi)\in (\mathbb R^n\setminus0)\times(\mathbb R^n\setminus0)$.
In your example, the operator $1-\Delta$ is SG elliptic, while the operator $-\Delta$ is not.
EDIT
If one is particularly interested in the operator $1-\Delta$, with $\Delta=\Delta_g$ and $g$ being an SG metric on $X$, then one immediately gets that $1-\Delta\colon H^2(X) \longrightarrow L^2(X)$ is an isomorphism (because $1-\Delta$ is a positive selfadjoint operator in $L^2(X)$ with domain $H^2(X)$), and then - employing the SG calculus - one concludes that $1-\Delta\colon W^{2,p}(X) \longrightarrow L^p(X)$ is an isomorphism for all $1<p<\infty$ (here, the spaces $W^{\sigma,p}(X)$ for $\sigma\in\mathbb R$, $1<p<\infty$ are defined with the help of the metric $g$).
Lockhard and McOwen actually use a different compactification of $\mathbb R^n$. They consider infinity as a conic point. In this case, the function spaces are different. When these are defined with the help of a cone metric $g$, then there is a discrete set of $p$ for which $1-\Delta\colon W^{2,p}(X) \longrightarrow L^p(X)$ is not even a Fredholm operator. Whenever one passes over such an exceptional $p$, the index for the remaining $p$ is going to change. A good reference can be found here.
