Assume a constant risk-free rate $r$ and no dividends. Generalisation is straightforward.
To preclude arbitrage opportunities, under the risk-neutral measure $\Bbb{Q}$, the discounted asset price process should be a $\Bbb{Q}$-martingale i.e.
$$ S_0 = \Bbb{E}^\Bbb{Q}_0 \left[ e^{-rt} S_t \right] \iff \Bbb{E}^\Bbb{Q}_0 \left[ S_t \right] = S_0 \exp(rt) \tag{1} $$
Now, rewriting your equation as
\begin{align}
S_t &= S_0 \exp(\alpha t) \mathcal{E}(\sigma W_t) \exp(-\beta \lambda t + J_t)
\end{align}
where $ J_t = \sum_{i=1}^{N_t} Y_i $ denotes a compound Poisson process with $\{Y_i\}_{i=1}^\infty$ i.i.d. random variables and $N_t$ a Poisson process of intensity $\lambda$, and taking the expectation under $\Bbb{Q}$ bearing in mind that the Wiener process is independent from the compound Poisson process yields
\begin{align}
\Bbb{E}_0^\Bbb{Q} [S_t] &= S_0 \exp(\alpha t) \exp(-\beta \lambda t ) \Bbb{E}_0^\Bbb Q[\exp(J_t)] \tag{2}
\end{align}
Comparing the above expressions, we see that $(1)$ is consistent with $(2)$ if and only if $ \alpha = r $ and $ \beta $ is such that
$ \Bbb{E}_0^\Bbb{Q}[ \exp(J_t) ] = \exp(\beta \lambda t ) $
Evaluating $\Bbb{E}_0^\Bbb{Q}[ \exp(J_t) ]$ gives
\begin{align}
\Bbb{E}_0^\Bbb{Q}[ \exp(J_t) ] &= \Bbb{E}_0^\Bbb{Q} \left[ \exp \left(\sum_{i=1}^{N_t} Y_i\right) \right] \\
&= \Bbb{E}_0^\Bbb{Q} \left[ \Bbb{E}_t^\Bbb{Q} \left[ \exp \left(\sum_{i=1}^{N_t} Y_i\right) \right] \right] \\
&= \sum_{n=0}^\infty \Bbb{E} \left[ \exp\left(\sum_{i=1}^n Y_i\right) \right] \Bbb{Q}(N_t = n) \\
&= \sum_{n=0}^\infty \prod_{i=1}^n \Bbb{E} \left[\exp(Y_i) \right] \Bbb{Q}(N_t = n) \\
&= \sum_{n=0}^\infty \left( \Bbb{E} \left[ \exp(Y_1) \right] \right)^n \Bbb{Q}(N_t = n) \\
&= e^{-\lambda t} \sum_{n=0}^\infty \left( \Bbb{E} \left[ \exp(Y_1) \right] \right)^n \frac{(\lambda t)^n}{n!} \\
&= \exp \left(\Bbb{E} \left[ \exp(Y_1) \right] - 1)\lambda t \right)
\end{align}
thereby showing that, under $\Bbb{Q}$, it is enough that $$ \alpha = r,\ \ \beta = \Bbb{E} \left[ \exp(Y_1) \right] - 1 $$
to preclude arbitrage opportunities.