Consider the cyclic group $\mathbb{Z}_n=\{0,1,\cdots,n-1\}$ (of order $n$). It is our conjecture that:
If $n\geq 8$ then there exist subsets $B,C$ such that $|B|\neq |C|$ and $$ \{0,1,n-1\}\cap (B-B)=\{0,1,n-1\}\cap (C-C)=\{0\}, $$ $$ \{0,1,n-1\}+B=\{0,1,n-1\}+C=\mathbb{Z}_n $$ (especially if $n$ is even).
Note that the property is not valid if $n<8$ and $A-A=\{a_1-a_2: a_1,a_2\in A\}$.
Extending the example for $n=6$, made in my comment, it is easy to find a general solution for $n\ge8$:
Take $C=\left\{0,2,4,6,\ldots,2\cdot(\lfloor n/2\rfloor - 1)\right\}$ and $B=(C\setminus\{2,4\})\cup\{3\}$. It is easy to verify that these sets satisfy all of your conditions.