I'm not familiar with your notation, but is $V$ the standard representation of $\text{SO}_{10}\mathbb{C}$? If so, $\text{Sym}^k V$ is the representation of $\text{GL}_{10}\mathbb{C}$ corresponding to the partition $\lambda=(k,0,...,0)$. Then you can apply Littlewood's formula to see how this irreducible decomposes when you restrict from $\text{GL}_{10}\mathbb{C}$ to $\text{SO}_{10}\mathbb{C}$: the multiplicity of the irreducible representation $V_{[\mu]}$ corresponding to the partition $\mu$ will be

$\sum_\eta C_{\eta\mu}^\lambda$

where $C_{\cdot\cdot}^\cdot$ is the Littlewood-Richardson coefficient, and the sum is over all partitions $\eta=(\eta_1,\eta_2,\eta_3,\eta_4,\eta_5)$ with each $\eta_i$ even. (Fulton-Harris "Representation theory", Equation 25.37, p. 427) Hopefully you can check whether this agrees with your expected answer.

I'm not familiar with your notation, but is $V$ the standard representation of $\text{SO}_{10}\mathbb{C}$? If so, $\text{Sym}^k V$ is the representation of $\text{GL}_{10}\mathbb{C}$ corresponding to the partition $\lambda=(k,0,...,0)$. Then you can apply Littlewood's formula to see how this irreducible decomposes when you restrict from $\text{GL}_{10}\mathbb{C}$ to $\text{SO}_{10}\mathbb{C}$: the multiplicity of the irreducible representation $V_{[\mu]}$ corresponding to the partition $\mu$ will be

$\sum_\eta C_{\eta\mu}^\lambda$

where $C_{\bullet\bullet}^\bullet$ is the Littlewood-Richardson coefficient, and the sum is over all partitions $\eta=(\eta_1,\eta_2,\eta_3,\eta_4,\eta_5)$ with each $\eta_i$ even. (Fulton-Harris "Representation theory", Equation 25.37, p. 427) Hopefully you can check whether this agrees with your expected answer.