Even though the fit seems correct, NonlinearModelFit throws a failed convergence error

I think there are few issues with what you're attempting to do:

• Fitting 6 parameters ($a$, $rho$, $phi$, $psi$, $r$, and the error variance) to just 12 data points is usually not enlightening. (This is not to say that obtaining more data is easy or even possible to do.)
• The parameters estimates for $rho$, $phi$, and $psi$ are not even close to being statistically significant.
• The correlation matrix estimate has nearly every correlation equal to 1. In other words, the model is way overparameterized given the available data. Predictions might be OK but you should definitely avoid interpreting coefficients (either the magnitude or the sign).

Here is the issue with the parameter estimates:

fit = NonlinearModelFit[data, -a + ρ r^m Cos[m ϕ + ψ], {a, ρ, ϕ, ψ, r}, m,
  MaxIterations -> 1000, Method -> "NMinimize"]
fit["ParameterTable"]

Here is the issue with the correlation matrix:

fit["CorrelationMatrix"] // MatrixForm 

The model is just overparameterized given the available data. Note that if one rationalizes the data, only the P-value for $a$ remains statistically significant:

fit2 = NonlinearModelFit[Rationalize[data, 0], {-a + ρ r^m Cos[m ϕ + ψ]}, {a, ρ, ϕ, ψ, r}, m, 
   MaxIterations -> 1000, Method -> "NMinimize", WorkingPrecision -> 30];
fit2["ParameterTable"]

fit2["CorrelationMatrix"] // MatrixForm