We present an explicit numerical approximation scheme, denoted by $\{X^n\}$, for the effective simulation of solutions $X$ to a multivariate stochastic differential equation (SDE) with a superlinearly growing $κ$-dissipative drift, where $κ>1$, driven by a multiplicative heavy-tailed Lévy process that has a finite $p$-th moment, with $p>0$. We show that for any $q\in (0,p+κ-1)$, the strong $L^q$-convergence $\sup_{t\in[0,T]}\mathbf{E} \|X^n_t-X_t\|^q=\mathcal{O} (h_n^γ)$ holds true, in particular, our numerical scheme preserves the $q$-moments of the solution beyond the order $p$. Additionally, for any $q\in (0,p)$ we establish strong uniform convergence: $\mathbf{E}\sup_{t\in[0,T]} \|X^n_t-X_t\|^q=\mathcal{O} ( h_n^{δ_q^\mathrm{uc}} )$. In both cases we determine the convergence rates $γ$ and $δ_q^\mathrm{uc}$. In the special case of SDEs driven solely by a Brownian motion, our numerical scheme preserves super-exponential moments of the solution. The scheme $\{X^n\}$ is realized as a combination of a well-known Euler method with a Lie--Trotter type splitting technique.