The statement
If $(X_k)$ is a sequence of independent r.v.'s uniformly bounded in absolute value by some real $c$ such that $\sum_{k}X_k$ converges a.s., then $\sum_{k}EX_k$ converges
follows almost immediately from the Hoffmann–Jørgensen inequality $$EM^2\le2.4^2c^2+32t_0^2$$ (see e.g. Proposition 6.8), where $M:=\sup_k|S_k|$, $S_k:=\sum_{j=1}^k X_j$, and $t_0>0$ is such that $P(M>t_0)<1/32$. Indeed, since $S_\infty$ is finite a.s., we have $M<\infty$ a.s. and hence we can assume that $t_0<\infty$, so that $Var S_n\le ES_n^2\le EM^2\le2.4^2c^2+32t_0^2<\infty$ for all natural $n$.
Now Kolmogorov's two-series theorem implies that $\sum_k(X_k-EX_k)$ converges a.s. Since $\sum_k X_k$ converges a.s., we conclude that $\sum_k EX_k$ converges, as desired.