Consider a pretty simple, log-linearized model. I understand that this question requires more background information but, in general, by how much can the estimates of policy and transition functions be different between Dynare results (using default options) and when solving the model by hand using the method of undetermined coefficients? Maybe they should even be identical? For example, I suppose that having estimates 0.63 and 0.75 for the same coefficient is more likely a signal that there is a mistake somewhere?

If there is a unique determinate equilibrium, the only difference can come from numerical imprecision. Any differences above 1e-4 should raise massive red flags.