We claim the following holds.
Proposition (cf. [Tan17, Prop. 3 and Idea of Thm. 1; Wan, Proof of Thm. 3.5, Case 2]). Let $X$ be a complete normal variety $X$ of dimension $d$ over an infinite perfect field of characteristic $p > 0$. Let $\Delta$ be an effective $\mathbf{Q}$-Weil divisor on $X$ such that the Cartier index of $K_X+\Delta$ is not divisible by $p$, and such that the pair $(X,\Delta)$ is sharply $F$-pure. Let $B$ be a free Cartier divisor. Then, there exists an effective $\mathbf{Q}$-Cartier divisor $B' \sim_{\mathbf{Q}} B$ such that
- the Cartier index of $K_X+\Delta+B'$ is not divisible by $p$;
- $(X,\Delta+B')$ is sharply $F$-pure; and
- the $F$-pure centers of $(X,\Delta)$ and those of $(X,\Delta+B')$ coincide.
The idea is to stratify $X$ into a disjoint union of minimal $F$-pure centers $W_i^k(X,\Delta)$. We can then apply the strategy in [Tan17] and [Wan] to show that we can find $B'$ such that $(W_i^k(X,\Delta),\Delta_{W_i^k(X,\Delta)}+B'\rvert_{W_i^k(X,\Delta)})$ is strongly $F$-regular for every $i$ and $k$, which implies the conditions (2) and (3) above by $F$-adjunction [Sch09]. I would like to make the disclaimer that the proof below is probably not optimal.
Proof. We first stratify the $F$-pure centers of $(X,\Delta)$. Let $W_i^*(X,\Delta)$ be the union of $F$-pure centers of $(X,\Delta)$ of dimension $\le i$, let
$$W_i(X,\Delta) := W_i^*(X,\Delta) \smallsetminus W_{i-1}^*(X,\Delta),$$
and let $W_i(X,\Delta) = \bigcup_k W_i^k(X,\Delta)$ be a decomposition into irreducible components. Since the intersection of $F$-pure centers is a union of $F$-pure centers of smaller dimension [Sch10, Lem. 3.5 and Prop. 4.7], we see that $X$ is the disjoint union
$$X = U \sqcup \bigsqcup_{i,k} W_i^k(X,\Delta),$$
where $(X,\Delta)$ is strongly $F$-regular on $U$. Moreover, denoting
$$Y_i^k(X,\Delta) := X \smallsetminus \bigl(\overline{W_i^k(X,\Delta)} \smallsetminus W_i^k(X,\Delta)\bigr),$$
we see that $W_i^k(X,\Delta)$ is a minimal $F$-pure center of $(Y_i^k(X,\Delta),\Delta\rvert_{Y_i^k(X,\Delta)})$ for every $i$ and $k$. In particular, $W_i^k(X,\Delta)$ is normal for every $i$ and $k$ by [Sch10, Cor. 7.8].
Next, for each $W^k_i(X,\Delta)$, [Sch09, (i) and (iv) of Main Thm.] implies that there exists an effective $\mathbf{Q}$-Weil divisor $\Delta_{W^k_i(X,\Delta)}$ on $W^k_i(X,\Delta)$ such that
$$(K_X+\Delta)\rvert_{W^k_i(X,\Delta)} = K_{W^k_i(X,\Delta)} + \Delta_{W^k_i(X,\Delta)},$$
and such that $(W^k_i(X,\Delta),\Delta_{W^k_i(X,\Delta)})$ is strongly $F$-regular.
We now prove it suffices to show the following:
Claim. There exists a nonempty open subset $V \subseteq \lvert B \rvert$ such that for all sufficiently large integers $e > 0$ for all $k$-rational $d$-tuples
$$(D_1,D_2,\ldots,D_d) \in V^{\times_k d} \subseteq \lvert B \rvert^{\times_k d},$$
the pairs
$$\biggl(U,\Delta\rvert_U+\frac{1}{p^e-1}(D_1+D_2+\cdots+D_d)\rvert_{U}\biggr)$$
and
$$\biggl(W_i^k(X,\Delta),\Delta_{W_i^k(X,\Delta)}+\frac{1}{p^e-1}(D_1+D_2+\cdots+D_d)\rvert_{W_i^k(X,\Delta)}\biggr)$$
are strongly $F$-regular for every $i$ and $k$.
Assuming the Claim, since $\lvert B \rvert$ is free, we can find
$$D_1,D_2,\ldots,D_{p^e-1} \in V$$
such that the intersection $\bigcap_{j \in J} D_j$ is empty for every subset $J \subseteq \{1,2,\ldots,p^e-1\}$ of cardinality $d+1$. For every point $x \in X$, there is an open neighborhood $T \subseteq X$ of $x$ such that
\begin{align*}
&\biggl(T \cap U,\Delta\rvert_{T \cap U}+\frac{1}{p^e-1}(D_1+D_2+\cdots+D_{p^e-1})\rvert_{T \cap U}\biggr)\\
&\qquad\qquad= \biggl(T \cap U,\Delta\rvert_{T \cap U}+\frac{1}{p^e-1}\sum_{i \in I} D_i\rvert_{T \cap U}\biggr)
\end{align*}
for a set of indicies $I$ with cardinality at most $d$ by the intersection condition on the $D_j$, and similarly
\begin{align*}
&\biggl(T \cap W_i^k(X,\Delta),\Delta_{W_i^k(X,\Delta)}\rvert_{T \cap W_i^k(X,\Delta)}+\frac{1}{p^e-1}(D_1+D_2+\cdots+D_{p^e-1})\rvert_{T \cap W_i^k(X,\Delta)}\biggr)\\
&\qquad\qquad=\biggl(T \cap W_i^k(X,\Delta),\Delta_{W_i^k(X,\Delta)}\rvert_{T \cap W_i^k(X,\Delta)}+\frac{1}{p^e-1}\sum_{i \in I_i^k} D_i\rvert_{T \cap W_i^k(X,\Delta)}\biggr)
\end{align*}
for sets of indices $I_i^k$ with cardinality at most $d$. These pairs are strongly $F$-regular by the choice of $V$ in the Claim. Finally, we note that setting
$$B' = \frac{1}{p^e-1}(D_1+D_2+\cdots+D_{p^e-1}),$$
condition (1) holds by construction, and conditions (2) and (3) hold by [Sch09, (v) of Main Thm.].
It remains to show the Claim. Consider the morphisms
$$W_i^k(X,\Delta) \times_k \lvert B \rvert^{\times_k d} \longrightarrow X \times_k \lvert B \rvert^{\times_k d} \overset{\mathrm{pr}_2}{\longrightarrow} \lvert B \rvert^{\times_k d}\tag{1}\label{eq:projectionmaps}$$
for every $i$ and $k$, where the first morphism is induced by the inclusion morphism $W_i^k(X,\Delta) \hookrightarrow X$. Choose an effective divisor $D' \in \lvert B \rvert$ containing no $F$-pure centers of $(X,\Delta)$ and an integer $e_0 > 0$ such that
$$\biggl(U,\Delta\rvert_U + \frac{d}{p^e-1}D'\rvert_U\biggr)$$
and
$$\biggl(W_i^k(X,\Delta),\Delta_{W_i^k(X,\Delta)} + \frac{d}{p^e-1}D'\rvert_{W_i^k(X,\Delta)}\biggr)$$
are strongly $F$-regular for all $i$ and $k$ and for every integer $e \ge e_0$. By applying [PSZ18, Cor. 4.19] to the morphisms in \eqref{eq:projectionmaps}, we see that there exists an open neighborhood $V \subseteq \lvert B \rvert$ of $D'$ such that the pairs
$$\biggl(U,\Delta\rvert_U + \frac{1}{p^e-1}(D_1+D_2+\cdots+D_d)\rvert_U\biggr)$$
and
$$\biggl(W_i^k(X,\Delta),\Delta_{W_i^k(X,\Delta)} + \frac{1}{p^e-1}(D_1+D_2+\cdots+D_d)\rvert_{W_i^k(X,\Delta)}\biggr)$$
are strongly $F$-regular for every $k$-rational $d$-tuple
$$(D_1,D_2,\ldots,D_d) \in V^{\times_k d},$$
every $i$ and $k$, and every integer $e \ge e_0$. $\blacksquare$
Remark. I would imagine that one can run a similar argument with the weaker assumption that the ground field $k$ contains an infinite perfect field of characteristic $p > 0$, using ideas from [Tan17, Prop. 2] instead of [PSZ18, Cor. 4.19].
References
[PSZ18] Zsolt Patakfalvi, Karl Schwede, and Wenliang Zhang. "$F$-singularities in families." Algebr. Geom. 5 (2018), no. 3, 264–327. DOI: 10.14231/AG-2018-009. MR: 3800355.
[Sch09] Karl Schwede. "$F$-adjunction." Algebra Number Theory, 3 (2009), no. 8, 907–950. DOI: 10.2140/ant.2009.3.907. MR: 2587408.
[Sch10] Karl Schwede. "Centers of $F$-purity." Math. Z., 265 (2010), no. 3, 687–714. DOI: 10.1007/s00209-009-0536-5. MR: 2644316.
[Tan17] Hiromu Tanaka. "Semiample perturbations for log canonical varieties over an $F$-finite field containing an infinite perfect field." Internat. J. Math. 28 (2017), no. 5, 1750030, 13 pp. DOI: 10.1142/S0129167X17500306. MR: 3655076.
[Wan] Yuan Wang. "On the characterization of abelian varieties for log pairs in zero and positive characteristic." Jan. 22, 2018. arXiv:1610.05630v2 [math.AG].