We revisit the binary adversarial wiretap channel (AWTC) of type II in which an active adversary can read a fraction $r$ and flip a fraction $p$ of codeword bits. The semantic-secrecy capacity of the AWTC II is partially known, where the best-known lower bound is non-constructive, proven via a random coding argument that uses a large number (that is exponential in blocklength $n$) of random bits to seed the random code. In this paper, we establish a new derandomization result in which we match the best-known lower bound of $1-H_2(p)-r$ where $H_2(\cdot)$ is the binary entropy function via a random code that uses a small seed of only $O(n^2)$ bits. Our random code construction is a novel application of pseudolinear codes -- a class of non-linear codes that have $k$-wise independent codewords when picked at random where $k$ is a design parameter. As the key technical tool in our analysis, we provide a soft-covering lemma in the flavor of Goldfeld, Cuff and Permuter (Trans. Inf. Theory 2016) that holds for random codes with $k$-wise independent codewords.