Combinatorial proofs of some identities on overpartitions with repeated smallest non-overlined part

Let $\overline{\mathrm{s}\mathrm{p}\mathrm{t}}k(n)$ denote the number of overpartitions of $n$ where the smallest non-overlined part, say $s(\pi )$, appears $k$ times and every overlined part is bigger than $s(\pi )$. Let $\overline{\mathrm{s}\mathrm{p}\mathrm{t}}{k}_o(n)$ denote the number of overpartitions of $n$ where the smallest non-overlined part appears $k$ times, every overlined part is bigger than $s(\pi )$ and all parts other than $s(\pi )$ are incongruent modulo $2$ with $s(\pi )$. Also, let $b_e(k,n)$ (resp., $b_o(k,n)$) denote the number of overpartitions of $n$ counted by $\overline{\mathrm{s}\mathrm{p}\mathrm{t}}{k}_o(n)$ where the number of parts greater than $s(\pi )$ is even (resp., odd), and let $$\overline{\mathrm{s}\mathrm{p}\mathrm{t}}{k}_o^{\mathrm{\prime }}(n)=b_e(k,n)-b_o(k,n)\mathrm{.}$$ Recently, Malik and Sarma (arXiv:2601.15601v1) expressed the generating functions of these partition functions in terms of linear combinations of $q$-series with polynomials in $q$ as coefficients. As corollaries, they derived some partition identities involving the functions for $k=1$ and sought for combinatorial proofs of their results. In this paper, we present some desired proofs.