The answer is still no and a variant of [my previous counter-example][1] will provide a counter-example for this new question.

Let $V = \mathbb{C}^4$ with basis $\{x,y,z,t\}$ and $\mathbb{C}^3 \subset V$ with basis $\{x,y,z\}$. Put $U = S^2 \mathbb{C}^3 \oplus \mathbb{C}.t^2$, a basis of which is given by $\{x^2, y^2, z^2, (x+y)^2, (x+z)^2, (y+z)^2, t^2\}$. This basis satisfies the hypothesis in the question as:
$$ \mathrm{span} \{x,y,z,x+y,x+z,y+z,t \} = V.$$

Note that $\dim U = 7$, $\dim S^2V = 10$ and that for any $W \subset V$ of dimension $2$ (so that $\dim S^2 W = 3$), we have:
$$S^2 \left(\mathbb{C}^3 \cap W \right) \subset \left(U \cap S^2W \right),$$
with $\dim \mathbb{C}^3 \cap W \geq 1$.

  [1]: https://mathoverflow.net/a/415807/37214