Scaling limit of the Ising model with nonzero order parameter

I'm interested in simulating the continuum limit of the 2D Ising model $$H=J\sum_{\langle i j\rangle} s_i s_j+ h \sum_i s_i$$

In one dimension I can fix average magnetization $$m=\langle s\rangle$$ and correlation length $$\xi=a\Delta i$$, $$a$$ being the lattice spacing, and take $$a\to 0$$. Because everything is exactly solvable, I can solve for $$\beta J$$ and $$\beta h$$ and expand about $$a=0$$ to find:

$$\beta J=-\frac{1}{2}\log(a)-\frac{1}{4}\log\left(\frac{1-m^2}{4\xi^2}\right)+O(a^2)$$ $$\beta h=a \frac{m}{2\xi}+O(a^3)$$

As I take $$a\to 0$$ I get a continuum theory with fixed correlation length and average magnetization. The logarithm is because the fixed point is at $$(\beta h,\beta J)=(0,\infty)$$.

In two dimensions, we have critical point $$(\beta h,\beta J)=(0,J_c)$$ and lots of great information given to us by critical exponent relationships like $$\xi\propto \tau^{-\nu}$$. But I haven't been able to figure out how to do the same thing.

How can I take the $$a\to 0$$ limit with fixed (not necessarily known) $$m$$ and $$\xi$$? Is this essentially the same question as asking how to take the $$\xi\to \infty$$ limit at fixed $$\langle s\rangle$$? I assume the relationships $$\xi\propto \tau^{-\nu}$$, $$m\propto(-\tau)^\beta$$, and $$h\propto m^\delta|_{\tau=0}$$ will be needed (where this takes the viewpoint that $$\xi\to\infty$$ instead of the equivalent $$a\to 0$$ picture).