Comparing model fits or regression coefficients for nonlinear models fitted to different data sets

I have kinetic data measured under several treatments (one experiment per treatment) which can be fitted by several nonlinear two-parameter models, how can I compare whether there there are differences in response that vary by treatment?

Comparing model fits. My initial thought was to use an F-test to compare relative residual sum-of-squares (RSS) across different data sets for each model, but if I understand correctly, this measures RSS relative to differences in degrees of freedom, so is more suitable for comparing a subset of a model relative to the full model for the same data set. Is there a similar test that would be appropriate in this case (to compare the same model applied to different data sets rather than different models with varying number of parameters to the same data set)?

Comparing regression coefficients. I found this reference which suggests to use a t-test where the test statistic is calculated by aggregating the standard errors of the nonlinear regression coefficients ($\hat{\beta}$ is the coefficient estimate and $\hat{s}_e$ is its standard error; subscripts 1 and 2 denote estimates for treatments 1 and 2):

$t = \frac{\hat{\beta}_1 - \hat{\beta}_2}{\sqrt{\hat{s}_{e,1}^2+\hat{s}_{e,2}^2}}$

with the combined degree of freedom being the sum of the original two. Is this a standard approach and is there a good literature/textbook reference?