The standard isoperimetric theorem inside the Euclidean space [71, Thm 6.1] does not state $\text{supp}(A^{\prime})\subset B_{R}$. Once we have found some $A^{\prime}$ with $\partial A^{\prime}=\partial A$ with mass control, we can pushforward by a Lipschitz retraction map $F:\mathbb{R}^{2n}\to B_{1}$:

$$F(x)=\begin{cases}x,\quad x\in B_{R},\\ \frac{Rx}{|x|},\quad x\in\mathbb{R}^{2n}\setminus B_{R}.\end{cases}$$

Replacing $A^{\prime}$ by $F_{*}A^{\prime}$, the mass cannot increase, and $\partial F_{*}(A^{\prime})=F_{*}(\partial A^{\prime})=F_{*}(\partial A)=\partial A$, and we have ensured the support is contained in $B_{R}$.