An element $g$ of $SU(2)$ is of the following form: $$ g=\begin{bmatrix} z_1 & z_2\\ -\bar{z}_2 & \bar{z}_1 \end{bmatrix}, $$ where $z_i$ are complex satisfying $|z_1|^2+|z_2|^2=1$. I can choose different parametrizations. The first one will be $z_1=|z_1|e^{i\phi},z_2=|z_2|e^{i\psi}$ in $\mathbb{C}^2$ used to derive a real parametrization \begin{align} x&=\cos{\phi}\cos{\beta}\\ y&=\sin{\phi}\cos{\beta}\\ z&=\cos{\psi}\sin{\beta}\\ t&=\sin{\psi}\sin{\beta} \end{align} for $0\leq\phi,\psi\leq2\pi,0\leq\beta\leq\pi/2$. This is called the Hopf map (see https://en.wikipedia.org/wiki/3-sphere) and the invariant measure (left and right since the group is compact) is $$d\mu=\sin{\beta}\cos{\beta}\ d\beta\ d\psi\ d\phi.$$
But I can use a different parametrization. I rewrite $g$ as $$ g=\begin{bmatrix} x_1+ix_2 & x_3+ix_4 \\ -x_3+ix_4 & x_1-ix_2 \end{bmatrix} $$ and introduce \begin{align} x_1&=\cos{\theta}\\ x_2&=\sin{\theta}\cos{\phi}\\ x_3&=\sin{\theta}\sin{\phi}\cos{\psi}\\ x_4&=\sin{\theta}\sin{\phi}\sin{\psi}. \end{align} The invariant measure is now $$d\nu=\sin^2{\theta}\ d\theta\sin{\phi}\ d\phi\ d \psi.$$
QUESTION: An invariant measure on a Lie compact group is unique up to a multiplicative factor. In what sense is $d\mu$ proportional to $d\nu$? I must have misunderstood the uniqueness claim.
Well, the same measure is obviously going to look different if you write it in different coordinates. For instance, Lebesgue measure on $\mathbb{R}^2$ is given by $dx\,dy$ in the usual coordinates but $r\,dr\,d\theta$ in polar coordinates. To compare them you would need to write them using the same coordinates.
Another way to say this is that you have described two maps $F,G:\mathbb{R}^3\to SU(2)$ and found two invariant measures $\mu$ and $\nu$ on $SU(2)$ for which $F^*\mu$ and $G^*\nu$ do not differ by a constant factor. But that shouldn't be a surprise, because it is $\mu$ and $\nu$ themselves which differ by a constant factor, not their pullbacks $F^*\mu$ and $G^*\nu$ in two different coordinate systems. What would be true is that $F^*\mu$ and $F^*\nu$ should differ by a constant factor--in other words, if you converted $d\nu$ into the same coordinates you used for your expression for $d\mu$, it would be the same up to a constant factor.