= Solution
Use $s=|t|$, $x=h/s^\Delta$ and separate scaling functions above and below the transition. Up to a nonsingular conversion between physical and dimensionless field, <magnetization scaling-function matching> gives
$$
m=-\partial_h f_s=s^{2-\alpha-\Delta}g_{m,\pm}(x),\qquad g_{m,\pm}=-g'_{f,\pm}.
$$
On the ordered side take $h\to0^+$ after the <thermodynamic limit>, selecting one <pure thermodynamic phase>. A finite nonzero $g_{m,-}(0^+)$ implies
$$
\boxed{\beta=2-\alpha-\Delta.}
$$
The critical isotherm must have a finite nonzero limit at fixed small $h$ as $s\to0$. Thus $g_m(x)\sim C\operatorname{sgn}(x)|x|^{1/\delta}$ at large $|x|$. Cancellation of the residual power of $s$ requires
$$
2-\alpha-\Delta-\frac\Delta\delta=0,\qquad
\boxed{\delta=\frac{\Delta}{2-\alpha-\Delta},\qquad\Delta=\beta\delta.}
$$
The absolute value and branch choice matter: spontaneous zero-field <magnetization> is below $T_c$, whereas the symmetry-unbroken <thermodynamic phase> has $m(t,0)=0$. The <field crossover exponent> is not the <scaling dimension> of the <magnetization> field.
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