Reference request concerning splitting fields for groups that are related to special symmetric groups

This is not an accurate answer but would be helpful. Take a look at chapter 2 of

Fields and Galois Theory By J.S. Milne that is available freely

and a quick review and lecture of KEITH CONRAD. There are some books on splitting fields like

Rotman, Joseph, Galois theory., Universitext. New York, NY: Springer. xiv, 157 p. (1998). ZBL0924.12001.

or

Lidl, Rudolf; Niederreiter, Harald, Introduction to finite fields and their applications., Cambridge: Univ. Press,. xi, 416 p. (1994). ZBL0820.11072.

or chapter 9 of

Roman, Steven, Field theory, Graduate Texts in Mathematics 158. New York, NY: Springer (ISBN 0-387-27677-7/hbk). xii, 332 p. (2006). ZBL1172.12001.

This condition you call being a splitting field is really just saying that all of the irreducible $\overline{\mathbb{F}}_p$ representations are defined over $\mathbb{F}_p$. For symmetric groups this is a standard fact, and can be found say in James' book.

Since the irreducible representations of $S_n$ are all defined over $\mathbb{F}_p$, so are the tensor products $V \otimes W$ which are the irreducible representations of $S_n \times S_n$.

For wreath products $S_n \wr C_2$ the irreducible representations in characteristic $p \ne 2$ are basically constructed the same way as characteristic zero: Each irreducible $S_k$ representation $D_\lambda$ can be extended in two ways to irreducible representations $D_\lambda^0$ and $D_\lambda^1$ of $S_k \wr C_2$ by declaring that each $C_2$ acts either trivially or by a sign. A general irreducible representation of $S_n \wr C_2$ is of the form $Ind_{S_k \wr C_2 \times S_{n-k} \wr C_2} (D_\lambda^0 \otimes D_\mu^1)$. Clearly these are defined over $\mathbb{F}_p$ since the $D_\lambda$'s are.