How does the functional measure transform under a field redefinition?

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.

Consider the more general case of a (not necessarily local) field redefintion of the form: begin{equation} phi(x)=int d y ,,f(x,y),gbig(theta(y)big)tag{1} end{equation} For example, in the question above we have begin{equation} phi(x)=int d y ,,delta(x-y),sinhbig(theta(y)big) end{equation} which is an example of a local field redefinition. In the discrete case where we think of the path integral as taking place on a lattice, Eq.(1) takes the form, where $phi_i$ is shorthand for the discrete variable $phi(x_i)$: begin{equation} phi_i=sum_j F_{ij}, g(theta_j) tag{2} end{equation} where $F_{ij}=F(x_i,x_j)$ can be thought of as a matrix and $F_{ij}=delta_{ij}$ corresponds to the case of a local transformation. Discretizing the path integral we can write the change of variables in Eq.(2) as (here $wedge$ denotes the wedge product, which is always present for the tensor density $d^dx$ but I make it explicit here only to make the presence of the Jacobian apparent) begin{align} int mathcal{D}phi&=int dphi_1wedge dphi_2wedge...wedge dphi_n &= int frac{1}{n!}epsilon_{i_1...i_n}dphi^{i_1}wedge...wedge dphi^{i_n} &=int frac{1}{n!}epsilon_{i_1...i_n},,frac{partialphi^{i_1}}{partialtheta^{i_1'}}...frac{partialphi^{i_1}}{partialtheta^{i_n'}}dtheta^{i_1'}wedge...wedge ,dtheta^{i_n'} &=int frac{1}{n!}epsilon_{i_1...i_n},,bigg(F^{i_1i_1'}frac{d g(xi)}{dxi}biggvert_{xi=theta_{i_1'}}bigg)...bigg(F^{i_ni_n'}frac{d g(xi)}{dxi}biggvert_{xi=theta_{i_n'}}bigg)dtheta^{i_1'}wedge...wedge ,dtheta^{i_n'} &=intfrac{1}{n!}epsilon_{i_1'...i_n'}text{Det}bigg[F^{ii'}frac{d g(xi)}{dxi}biggvert_{xi=theta_{i'}}bigg]dtheta^{i_1'}wedge...wedge ,dtheta^{i_n'} &=inttext{Det}bigg[F^{ii'}frac{d g(xi)}{dxi}biggvert_{xi=theta_{i'}}bigg]dtheta^{i_1'}wedge...wedge ,dtheta^{i_n'} &=int dtheta^{i_1'}wedge...wedge ,dtheta^{i_n'},,expbigglbracetext{Tr}bigg(logbigg[F^{ii'}frac{d g(xi)}{dxi}biggvert_{xi=theta_{i'}}bigg]bigg)biggrbracetag{3} end{align} So for example if we have a local field redefintion $F_{ij}=delta_{ij}$ then we encounter the term $Tr(log(delta_{ij}))=log(n)$, where $n$ is the number of lattice sites. In the continuous case where $F_{ij}rightarrow f(x,y)=delta^{d}(x-y)$ we encounter the highly singular term begin{equation} Tr(log(delta^d(x-y)))=int d^dx,,logbigg(delta^{d}(x-x)bigg)=int d^dx,,logbigg(delta^{d}(0)bigg) end{equation} So to answer the original question, I think the measure transforms as: begin{align} mathcal{D}phi=mathcal{D}theta,,expbigg(int dx,logbig(delta(0)big)+logbig(1+3ctheta^2)big)bigg)tag{1} mathcal{D}phi=mathcal{D}theta,,expbigg(int dx,logbig(delta(0)big)+logbig(3ctheta^2big)bigg)tag{2} mathcal{D}phi=mathcal{D}theta,,expbigg(int dx,logbig(delta(0)big)+logbig(cosh(theta(x))big)bigg)tag{3} end{align} Apparently one can ignore the $delta^d(0)$ when working in dimensional regularization as we can interpret $delta^d(0)$ as the volume of spacetime, which in $d-epsilon$ dimensions is begin{equation} delta^{d}(0)=frac{2pi^{d/2}}{Gamma(d/2)}frac{Gamma(-d)}{Gamma(1-d)} end{equation} which is $frac{1}{epsilon}$ dependent and hence we can use $frac{1}{epsilon}$ dependent counterterms to get rid of this term. However $exp(log(g'))$ terms still remain and it is unclear to me how to show that correlation functions will be unaffected by this term.