Why is the Power Method giving me one of the smaller eigenvalues instead of the dominant one?

Probably your second choice of $b_0$ was perpendicular to the eigenspace of the dominant eigenvalue.

For example, let $$A=\begin{pmatrix}3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1\end{pmatrix}.$$ Then, see that for any $\varepsilon>0$ we can define $$u=\begin{pmatrix}\varepsilon \\ 1 \\ 0\end{pmatrix}$$ and $$v=\begin{pmatrix}0 \\ 1 \\ 0\end{pmatrix}.$$

$u$ and $v$ are as close as we like, but $$A^n u = \begin{pmatrix}\varepsilon 3^n \\ 2^n \\ 0\end{pmatrix}$$ and $$A^n v = \begin{pmatrix} 0 \\ 2^n \\ 0\end{pmatrix}.$$

For large $n$, this coordinate $\varepsilon 3^n$ will dominate, but it vanished in $v$.