I hope to compute a functional integral $Z=\int \mathcal{D}\phi\,\, e^{-S[\phi]}$ with an action

$$S[\phi]=\int d^2x \sqrt{g}\Big((\nabla \phi)^2+\frac{1}{g}M^2(x) \phi^2\Big)$$

The scalar field $\phi$ is defined on a two-dimensional curved sphere.. The mass-like term $\frac{1}{g}M^2(x) \phi^2$ depends explicitly on $x$ and I'm interested in limit $g\to 0$. Formally the result is the functional determinant $\log Z=\log \operatorname{det} \Big(-\Delta+\frac{1}{g} M^2(x)\Big)$ and I'm interested the small $g$ expansion as a functional of $M^2(x)$.

I'm not very familiar with functional determinants but I've tried to apply the heat kernel method here without much success. The small $g$ expansion here does not seem to reduce to the conventional large mass expansion. Moreover the heat kernel coefficients have the form like $a_2=\int d^2x\sqrt{g}\Big(\frac16R-\frac{1}{g}M^2(x)\Big)$ while I naively expect that the leading order in small $g$ limit should be

$$\log \operatorname{det} \Big(-\Delta+\frac{1}{g} M^2(x)\Big)\sim \log \operatorname{det} \Big(\frac{1}{g} M^2(x)\Big)=\operatorname{Tr}\log \Big(\frac{1}{g} M^2(x)\Big)\sim \\\int d^2z \sqrt{g} \log\Big(\frac{1}{g} M^2(x)\Big)$$

Where the last line is my guess for what the functional trace of a function (diff operator of order 0) should be. Heat kernel method does not seem to produce logarithms like that.

Any comments and pointers to the literature are welcome.