To my mind, this can be done by a simple parametrix patching argument (I'm mimicking here the proof of 14.2.1 in Kronheimer-Mrowka's *Monopoles and Three-Manifolds*, but I am sure it was known much earlier).

Embed $X \setminus (3,\infty) \times Y$ into some closed $4$-manifold $Z$, isometrically. The Laplacian $L_Z:L_2^{p}(Z) \to L^{p}(Z)$ is Fredholm by elliptic theory. Take a parametrix $P_Z:L^p(Z) \to L_2^{p}(Z)$, i.e. an operator such that $P_Z L_Z - 1$ and $L_ZP_Z-1$ are compact. Similarly, for the Laplacian $L_{t}:L_2^{p,w}(\mathbb{R}\times Y) \to L^{p,w}(\mathbb{R} \times Y)$ one can onlo find $P_{t}$ such that $P_{t} L_{t}-1$ and $L_{t}P_{t}-1$ are compact.

Take $\beta_t, \beta_Z$ to be a smooth partition of unity subordinate to the covering of $X$ by $(1,\infty) \times Y$ and $X \setminus [2,\infty) \times Y$. Let $\eta_Z$ be a smooth function on $Z$ which is $1$ on the support of $\beta_Z$ and has support in $X \setminus [3,\infty) \times Y$; let $\eta_t$ be a smooth function on $\mathbb{R} \times Y$ which is $1$ on the support of $\beta_t$ and is supported in $(0,\infty) \times Y$.
This way, any function $f$ on $X$ can be expressed as the sum of $\beta_Z f$ and $\beta_t f$, which extend by $0$ to functions on $Z$ and $\mathbb{R} \times Y$ respectively. Vice versa, functions $g$, $h$ on $Z$, $\mathbb{R} \times Y$ respectively may be "glued together" to form a function on $X$ given by $\eta_Z g + \eta_t h$.

Now, we construct $P:L^{p,w}(X) \to L_2^{p,w}(X)$ via
$$ P(u) = \eta_Z P_Z (\beta_Z u) + \eta_t P_t(\beta_t u).$$

It is straightforward to check that $LP-1$ and $PL-1$ are compact.
For instance, in
$$LP(u) = L \eta_Z P_Z (\beta_Z u) + L \eta_t P_t (\beta_t u),$$
the first term can be expressed as
$$\eta_Z L P_Z (\beta_Z u) + K(u)$$
for some compact operator $K$ since derivatives of $\eta_Z$ have compact support and are smooth, and the inclusions $L_i^{p} \to L^{p}$ are compact for $i=1,2$ when restricted to compact subsets (to be precise, submanifolds of codimension $0$).
However, $\eta_Z L P_Z (\beta_Z u) = \eta_Z L_Z P_Z(\beta_Z u) = \eta_Z \beta_Z u + K'(u) = \beta_Z u + K'(u)$ where $K'$ is some compact operator, and one sees that $LP(u) = u + \tilde K(u)$ for some compact $\tilde K$.

The compactness of $PL-1$ is proven almost identically.