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We study the numerical approximation of the stochastic heat equation with a distributional reaction term. Under a condition on the Besov regularity of the reaction term, it was proven recently that a strong solution exists and is unique in the pathwise sense, in a class of H\"older continuous processes. For a suitable choice of sequence $(b^k)_{k\in \mathbb{N}}$ approximating $b$, we prove that the error between the solution $u$ of the SPDE with reaction term $b$ and its tamed Euler finite-difference scheme with mollified drift $b^k$, converges to $0$ in $L^m(\Omega)$ with a rate that depends on the Besov regularity of $b$. In particular, one can consider two interesting cases: first, even when $b$ is only a (finite) measure, a rate of convergence is obtained. On the other hand, when $b$ is a bounded measurable function, the (almost) optimal rate of convergence $(\frac{1}{2}-\varepsilon)$-in space and $(\frac{1}{4}-\varepsilon)$-in time is achieved. Stochastic sewing techniques are used in the proofs, in particular to deduce new regularising properties of the discrete Ornstein-Uhlenbeck process.