As far as I know, information criteria such as the AIC or BIC can only be used in combination with a likelihood. I can't tell how your fits are being performed, but you mentioning R^2 smacks of least squares.

Note that R^2 is only appropriate for linear models. See e.g. Article An evaluation of R2 as an inadequate measure for nonlinear m...

Maybe this thread can help you further with your problem:

https://www.researchgate.net/post/How-to-assess-goodness-of-fit-for-a-non-linear-model

Good luck!