No. If $\varphi \in C^\infty_c(B)$ is a bump function equal to $1$ in $|x| \leq 1/2$ then from Greens' theorem we have $$ \int_B u \Delta \varphi = 0$$ for all $u \in S_B$, but the same is not true in general for typical $u \in S_U$, which by the Cauchy-Schwarz inequality implies that $u$ is a positive distance away from $S_B$ in the $L^2(U)$ norm. For instance, if we take $u = K|_U \in S_U$ where $K(x) = \frac{-1}{4\pi |x|}$ is the Newton potential (the fundamental solution to $\Delta K = \delta$) then $$ \int_B u \Delta \varphi = \int (\Delta K) \varphi = 1 \neq 0$$ and hence $u$ is a positive distance from $S_B$.

One can create similar obstructions using functions $\varphi$ which behave like a specified spherical harmonic in the angular variable (instead of being constant in the angular variable, which is basically what is being done here).

No. If $\varphi \in C^\infty_c(B)$ is a bump function equal to $1$ in $\lvert x\rvert \leq 1/2$ then from Green's theorem we have $$ \int_B u \Delta \varphi = 0$$ for all $u \in S_B$, but the same is not true in general for typical $u \in S_U$, which by the Cauchy–Schwarz inequality implies that $u$ is a positive distance away from $S_B$ in the $L^2(U)$ norm. For instance, if we take $u = K\rvert_U \in S_U$ where $K(x) = \frac{-1}{4\pi \lvert x\rvert}$ is the Newton potential (the fundamental solution to $\Delta K = \delta$) then $$ \int_B u \Delta \varphi = \int (\Delta K) \varphi = 1 \neq 0$$ and hence $u$ is a positive distance from $S_B$.

One can create similar obstructions using functions $\varphi$ which behave like a specified spherical harmonic in the angular variable (instead of being constant in the angular variable, which is basically what is being done here).

No. If $\varphi \in C^\infty_c(B)$ is a bump function equal to $1$ in $|x| \leq 1/2$ then from Greens' theorem we have $$ \int_B u \Delta \varphi = 0$$ for all $u \in S_B$, but the same is not true in general for typical $u \in S_U$, which by the Cauchy-Schwarz inequality implies that $u$ is a positive distance away from $S_B$ in the $L^2(U)$ norm. For instance, if we take $u = K|_U \in S_U$ where $K(x) = \frac{-1}{4\pi |x|}$ is the Newton potential (the fundamental solution to $\Delta K = \delta$) then $$ \int_B u \Delta \varphi = \int (\Delta K) \varphi = 1 \neq 0$$ and hence $u$ is a positive distance from $S_U$.

One can create similar obstructions using functions $\varphi$ which behave like a specified spherical harmonic in the angular variable (instead of being constant in the angular variable, which is basically what is being done here).

No. If $\varphi \in C^\infty_c(B)$ is a bump function equal to $1$ in $|x| \leq 1/2$ then from Greens' theorem we have $$ \int_B u \Delta \varphi = 0$$ for all $u \in S_B$, but the same is not true in general for typical $u \in S_U$, which by the Cauchy-Schwarz inequality implies that $u$ is a positive distance away from $S_B$ in the $L^2(U)$ norm. For instance, if we take $u = K|_U \in S_U$ where $K(x) = \frac{-1}{4\pi |x|}$ is the Newton potential (the fundamental solution to $\Delta K = \delta$) then $$ \int_B u \Delta \varphi = \int (\Delta K) \varphi = 1 \neq 0$$ and hence $u$ is a positive distance from $S_B$.

One can create similar obstructions using functions $\varphi$ which behave like a specified spherical harmonic in the angular variable (instead of being constant in the angular variable, which is basically what is being done here).