How does the functional measure transform under a field redefinition?

1. It seems natural to generalize OP's setting to several fields $\phi^{\alpha}$ in $d$ spacetime dimensions. Under ultra-local field redefinitions$^1$ $$\begin{align} \phi^{\prime\alpha}(x)~=~&F^{\alpha}(\phi(x),x)\cr ~=~&\phi^{\alpha}(x)-f^{\alpha}(\phi(x),x),\end{align}\tag{1} $$ the Jacobian functional determinant in the path/functional integral is formally given as $$\begin{align} J~=~&{\rm Det} (\mathbb{M})\cr ~=~&\exp {\rm Tr}\ln (\mathbb{M})\cr ~=~& \exp\left(-\sum_{j=1}^{\infty} \frac{1}{j}{\rm Tr} (\mathbb{m}^j)\right) \cr ~=~& \exp\left(\delta^d(0) \int\! d^dx ~{\rm tr} (\ln M(x))\right),\end{align} \tag{2} $$ where we have defined $$\begin{align} \mathbb{M}~\equiv~&\mathbb{1}-\mathbb{m},\cr \mathbb{M}^{\beta}{}_{\alpha}(x^{\prime},x) ~:=~&\frac{\delta F^{\beta}(x^{\prime})}{\delta\phi^{\alpha}(x)}\cr ~=~& M^{\beta}{}_{\alpha}(x^{\prime})\delta^d(x^{\prime}\!-\!x),\cr M^{\beta}{}_{\alpha}(x)~:=~& \frac{\partial F^{\beta}(x)}{\partial\phi^{\alpha}(x)}~=~\delta^{\beta}_{\alpha}-m^{\beta}{}_{\alpha}(x),\cr \mathbb{m}^{\beta}{}_{\alpha}(x^{\prime},x) ~:=~&\frac{\delta f^{\beta}(x^{\prime})}{\delta\phi^{\alpha}(x)}\cr ~=~& m^{\beta}{}_{\alpha}(x^{\prime})\delta^d(x^{\prime}\!-\!x),\cr m^{\beta}{}_{\alpha}(x)~:=~& \frac{\partial f^{\beta}(x)}{\partial\phi^{\alpha}(x)}.\end{align} \tag{3}$$

2. If we discretize spacetime, then the Jacobian becomes a product of ordinary determinants $$ J~=~\prod_i \det (M(x_i)), \tag{4}$$ where the index $i$ labels lattice points $x_i$ of spacetime. The Dirac delta at zero $\delta^d(0)$ is here replaced by a reciprocal volume of a unit cell of the spacetime lattice, which can viewed as a UV regulator, cf. e.g. my Phys.SE answer here.

3. In dimensional regularization (DR), the Dirac delta at zero $\delta^d(0)$ vanishes, cf. Refs. 1 - 3. Heuristically, DR only picks up residues of various finite parameters of the physical system, while contributions from infinite parameters are regularized to zero. As a consequence, in DR the Jacobian $J=1$ becomes one under local field redefinitions (if there are no anomalies present).

References:

1. M. Henneaux & C. Teitelboim, Quantization of Gauge Systems, 1994; Subsection 18.2.4.

2. G. Leibbrandt, Introduction to the technique of dimensional regularization, Rev. Mod. Phys. 47 (1975) 849; Subsection IV.B.3 p. 864.

3. A.V. Manohar, Introduction to Effective Field Theories, arXiv:1804.05863; p. 33-34 & p. 51.

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$^1$ Much of this can be generalized to local field redefinitions $$ \begin{align}\phi^{\prime\alpha}(x)~=~&F^{\alpha}(\phi(x),\partial\phi(x),\partial^2\phi(x), \ldots ,\partial^N\phi(x) ,x)\cr ~=~&\phi^{\alpha}(x)\cr ~-~&f^{\alpha}(\phi(x),\partial\phi(x),\partial^2\phi(x), \ldots ,\partial^N\phi(x) ,x),\end{align}\tag{5}$$ and derivatives $\partial^j\delta^d(0)$ of the Dirac delta at zero. A local field redefinition corresponds to insertion of UV-relevant/IR-irrelevant vertices in the action, i.e. it doesn't change low-energy physics.

All three cases, (1)-(3), are local redefinitions, meaning that the value of $\phi(x)$ for any given $x$ is determined only by the value of $\theta(x)$ at that same value of $x$ (and conversely, assuming it's invertible).

Conceptually, the parameter $x$ is just a continuous index labeling different integration variables. In fact, the most generally-applicable way we have for defining a functional integral (at least in QFT) is to replace this continuous parameter with a discrete index. Then you have an ordinary multi-variable integral, and the rule for changing integration variables is the usual one. So the cases (1)-(3) just describe changes-of-variable in a bunch of single-variable integrals.

Thinking about things this way (with $x$ discretized) should help track down what's really going on with the $\delta(x-x')$ factor.