Based on comments from Eric Rowell and José Figueroa-O'Farrill, I assume that your $V$ is a half-spinor representation of $Spin_{10}$. The key issue is that your hypothetical decomposition of $S(V)$ is *multiplicity-free*, i.e. any module that appears in it has multiplicity one. There is a beautiful theory of multiplicity-free actions that allows you to do the following.

(1) Test that $V$ is a multiplicity-free $G$-module.

(2) Find the highest weights of the simple summands in $S(V)$.

For (1), the action is multiplicity-free if and only if the Borel subgroup $B$ has an open orbit on $V$. For (2), the highest weights form a free commutative semigroup whose generators are determined from the stabilizer in $B$ of a point in the open orbit.

An excellent survey is Roger Howe's *Schur lectures.*

All multiplicity-free linear actions (i.e. modules) of reductive algebraic groups have been classified. A convenient reference is

Howe and Umeda, *The Capelli identity, the double commutant
theorem and multiplicity-free-actions*, Mathematische Annalen, 290, 565 - 620 (freely available through GDZ, but tricky to link to).

Item (x) of the list (11.0.1) on p.583 is your action $Spin_{10}\times GL_1.$ This module arises from the action of a Levi factor of a maximal parabolic of $E_6$ on its abelian nilradical (remove last node from the Dynkin diagram of $E_6$ to get $D_5$). The decomposition is described in 11.10 on p.602: there are two fundamental heighest weights, one in degree 1 corresponding to the Spin module itself and one in degree 2 corresponding to the standard 10-dimensional representation of $SO_{10}.$ This leads to the decomposition you have stated.