Why could it be assumed that the standard enthalpy change is independent of temperature down to 25 °C?
Generally speaking you can write that 
$$\Delta H^\circ(T) = \Delta H^\circ(T_\mathrm{ref}) + \int_{T_\mathrm{ref}}^T \Delta C_p \,\mathrm dT \tag{1}$$
where $T_\mathrm{ref}$ might be $\pu{25 ^\circ C}$ and the pressure is 1 bar (in your case, approximately 1 bar). 
If your temperature window is not too large you may get away with assuming $\Delta H^\circ$ is constant. This amounts to assuming that the second term in the equation, dependent on $\Delta C_p$, is small compared to the first. If it isn't then rather than using the integrated van't Hoff equation 

$$\ln\left(\frac{K_2}{K_1}\right)=\frac{-\Delta H^\circ}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right) \label{eqn:2}\tag{2}$$
maybe you should attempt a polynomial fit to $\ln K_\mathrm{eq}$ and taking the derivative of the polynomial wrt $T^{-1}$ to evaluate $\Delta H^\circ$ as a function of $T$.

To test whether \eqref{eqn:2} is appropriate you might evaluate the quality of a fit to that equation or check the linearity of a plot of $\ln K$ versus $T^{-1}$.