Suppose $\vec{u}, \vec{v} \in \left(V, \mathbb{C}^{n}\right)$: by construction $$ \begin{split} \vec{u} &= \sum_{i=1}^{n}u_{i}\vec{e}_{i}\\ \vec{v} &= \sum_{i=1}^{n}v_{i}\vec{e}_{i} \end{split}. $$
The inner product of vectors $\vec{u}$ and $\vec{v}$ gives
$$ \langle\ \vec{u}, \vec{v}\rangle = \langle \sum_{i=1}^{n}u_{i}\vec{e}_{i}, \sum_{i=1}^{n}v_{i}\vec{e}_{i}\rangle. $$
But $\langle\ \vec{u}, \vec{v}\rangle = \langle\ \vec{v}, \vec{u}\rangle^{*}$ with $*$ being the complex conjugate, and recalling that $\langle\ X,\alpha Y\rangle = \alpha^{*} \langle\ X,Y\rangle$ for $\alpha$ a complex scalar, this gives $$ \left(\sum_{i,j=1}^{n}u_{i}^{*}v_{j}\langle\ \vec{e}_{j}, \vec{e}_{i}\rangle \right)^{*} =\sum_{i,j=1}^{n}u_{i}v_{j}^{*} \langle\ \vec{e}_{j}, \vec{e}_{i} \rangle. $$
This is in contrast to the following
At which point is my argument flawed? Any help is appreciated.
You must take care that there are two different conventions:
In some cases the hermitian inner product is defined as $\langle u,v \rangle=u^t\bar{v}$,
in other cases it is defined as $\langle u,v \rangle=\bar{u}^t v=u^*v$ (mostly in quantum mechanics, Dirac's bra-ket notations)
Further details Alternative definitions, notations and remarks
It is the origin of the confusion, in your first statement you use the first convention, in the last one, the second one.