Gauge anomaly in Polyakov string and Faddeev-Popov method

I am currently trying to gain a better understanding of the gauge fixing procedure used in chapter 5 of David Tong's notes.

Since the central charge of the Polyakov action for, say, the bosonic string is not zero the measure $$\mathcal{D}X$$ has non-trivial dependence on $$g$$ in $$Z[g]=\int\mathcal{D}X\exp(-S_{Pol}[X,g])$$ where $$g$$ is the metric on the string (worldsheet). Importantly it changes non-trivially under supposed gauge transformations of $$g$$. I will from now on denote it as $$\mathcal{D}_gX$$.

Now, when gauge fixing using the Faddeev-Popov method to calculate

$$Z=\frac{1}{\mathrm{Vol}} \int\mathcal{D}g\int\mathcal{D}_gX\exp(-S_{Pol}[X,g]).\tag{p.109}$$ Tong inserts $$1=\Delta_{FP}[g]\int\mathcal{D}\zeta\delta(g-g_0^\zeta)\tag{5.1}$$ where $$\int\mathcal{D}\zeta$$ is an integral over the gauge group and $$g_0^\zeta$$ is a 'reference' metric acted on by the gauge group element $$\zeta$$. One gets, by integrating in $$g$$ using the delta function, $$Z=\frac{1}{\mathrm{Vol}}\int\mathcal{D}\zeta\int\mathcal{D}_{g_0^\zeta}X\Delta_{FP}[g_0^\zeta]\exp(-S_{Pol}[X,g_0^\zeta]) .\tag{p.111}$$ At this point Tong changes all $$g_0^\zeta$$ to $$g_0$$ and uses $$\frac{1}{\mathrm{Vol}}\int\mathcal{D}\zeta=1$$.

While the Faddeev-Popov determinant and the Polyakov action are really gauge invariant, the integration measure clearly isn't. Why is this not mentioned or acknowledged in the text? Am I having a misunderstanding of the nature of the conformal anomaly?