I would avoid the question altogether as secondary. The issue is the choice between two models, a linear fit and an exponential approach to an asymptote. The linear model may possibly be an acceptable fit depending partly on the range of your data, but a negative intercept implies negative productivity for zero rainfall and a positive intercept implies positive; the first is impossible and the second implausible; nevertheless the model might be tolerable with warnings.
Conversely, the exponential is evidently chosen to have the limiting behaviour of approaching the origin, but how well it fits in practice depends on whether the idea of an asymptote is correct and (simply, but crucially) on the impact of numerous other controls on productivity.
The central question of which model fits better should be clear from plotting data and model fits. Plotting residuals as well would do no harm.
There is scope for disagreement here, but I prefer to approach $R^2$ as the square of the correlation between observed and predicted. This is totally consistent with its use in linear regression and relatively easy to explain to non-statistical users. Either way, be aware that outside linear regression the many different definitions of $R^2$ may not agree.
There is some minor confusion in your question. Whether people use any kind of $R^2$, AIC or BIC does not really stamp them as frequentist or Bayesian. There could be qualifications and details attached to that, but arguments over frequentist and Bayesian approaches to inference arguably have no bearing on your question.
(LATER) It has become apparent that the observations are points on two curves, productivity so far this season and rainfall so far this season. I suppose this was implicit in the term "growth curve" but it was not obvious to me and causes severe qualifications to my original comments.
Cumulation imparts a dependence structure which renders any inferential machinery for $R^2$, AIC and BIC invalid, unless somehow calculations took the cumulation into account. Thus all inferential bets are off unless the cumulation is corrected for in the model assessment. (I've seen allusion to this problem of correlating cumulative curves in the glaciology literature.)
I was supposing that (rainfall, productivity) pairs for each location were from different seasons, which still leaves the possibility of season to season dependence, which in practice is likely to be weaker. To spell out the point, it seems that each location is being characterised by data from one season.
@Armel expresses disagreement with my statement that a curve of the form
$ a\cdot(1-\exp[-b\cdot\text{Rain}(t)])$ can not be S-shaped, but does not give any reasons. I am referring to the shape of the curve on a plot with Rain on one axis; the point then is one of elementary calculus that such a curve has no inflexion. Here what I understand by S-shaped is that an inflexion exists at which curvature changes sign. (The relationship between productivity and time could be much more complicated geometrically, depending on the dependence of cumulative rainfall on time.)
Best Answer
It is possible indeed. As explained at https://methodology.psu.edu/AIC-vs-BIC, "BIC penalizes model complexity more heavily. The only way they should disagree is when AIC chooses a larger model than BIC."
If your goal is to identify a good predictive model, you should use the AIC. If your goal is to identify a good explanatory model, you should use the BIC. Rob Hyndman nicely summarizes this recommendation at
https://robjhyndman.com/hyndsight/to-explain-or-predict/:
"The AIC is better suited to model selection for prediction as it is asymptotically equivalent to leave-one-out cross-validation in regression, or one-step-cross-validation in time series. On the other hand, it might be argued that the BIC is better suited to model selection for explanation, as it is consistent."
The recommendation comes from Galit Shmueli’s paper “To explain or to predict?”, Statistical Science, 25(3), 289-310 (https://projecteuclid.org/euclid.ss/1294167961).
Addendum:
There is a third type of modeling - descriptive modeling - but I don't know of any references on which of AIC or BIC is best suited for identifying an optimal descriptive model. I hope others here can chime in with their insights.