Goodness of Fit. Perhaps not exactly as written by the OP, but it looks to me like an appropriate test. Perhaps there is another that you have in mind, or could you clarify why you do not think it applies here?
Another way to assess the goodness of fit would be to test for the "lack of fit" i.e. test for the deviations about the non-linear model.
You don't say whether you have replicate observations or not. However, you can construct a lack of fit test if you have replicate observations for your x variable.
1) You can estimate the pure error sums of square (SS_pure) by fitting a completely randomised design (i.e. x as a class variable).
2) You can obtain residual sums of square from your non-linear model (SS_nlin).
3) Then Your lack of fit SS will be the difference between 1 and 2.
4) Finally you can partition the residual SS from your non-linear model into pure error and lack of fit and constuct your F test to judge the lack of fit of your non-linear model.