Yes, I think you can extract Raynaud's result straightforwardly from the Breuil--Kisin theory. Write $k$ for the residue field of $R$. Let's recall how the Breuil--Kisin theory works. Let $\phi : k[[u]] \to k[[u]]$ be the $p$th power map. Then there's an anti-equivalence between the category of $p$-torsion finite flat group schemes over $R$, and the category of finite rank free $k[[u]]$-modules $\mathfrak{M}$ equipped with a $\phi$-semilinear map $\varphi : \mathfrak{M} \to \mathfrak{M}$ such that the image of the ($k[[u]]$-linear) map $\varphi^* : \varphi^* \mathfrak{M} := k[[u]] \otimes_{\phi, k[[u]]} \mathfrak{M} \to \mathfrak{M}$ contains $u^e \mathfrak{M}$. Finally let's recall that the operation of passing to the generic fibre of the group scheme is recovered on the Breuil--Kisin module side by passing from $\mathfrak{M}$ to the étale $\phi$-module $\mathfrak{M}[1/u]$.

So, suppose you have Breuil--Kisin modules $\mathfrak{M} \supset \mathfrak{N}$ with $\mathfrak{M}[1/u] = \mathfrak{N}[1/u]$ (one can always reduce to this case). Choose the least integer $r \ge 0$ such that $\mathfrak{N} \supset u^r \mathfrak{M}$, and take $m \in \mathfrak{M}$ with $n := u^r m \in \mathfrak{N}, u^{r-1} m \not\in \mathfrak{N}$. Since $\varphi(m) \in \mathfrak{M}$ we in particular have $u^r \varphi(m) \in \mathfrak{N}$, and so we require $u^{e+r}\varphi(m)$ to be in the image of $\varphi^*\mathfrak{N}$ under the map $\varphi^*$.

But $u^{e+r}\varphi(m) = \varphi^*(u^{e+r} \otimes_\phi m) = \varphi^*(u^{e-(p-1)r} \otimes_\phi n)$. Since $\varphi^*$ is an isomorphism after inverting $u$, we deduce that $u^{e-(p-1)r} \otimes_\phi n$ must lie in $\varphi^*\mathfrak{N}$, and therefore $e \ge (p-1)r$. If in particular $e < p-1$ then $r=0$ and $\mathfrak{M}=\mathfrak{N}$.

vastlymore difficult foundation than Raynaud's proof, so it wouldn't be "simple" in the end compared to Raynaud's proof (but would give another perspective, to be sure). That isn't to say it isn't an instructive example to think through, but (as you may know) the real significance of the B-K formalism in practice is with the cases $e \ge p$. Have you tried to do it yourself using the description of Raynaud's "$F$-vector schemes of rank 1" via Breuil modules in section 2 of math.jhu.edu/~savitt/papers/pdfs/appendix-final.pdf? $\endgroup$