@AbdelmalekAbdesselam this is exactly my question (in a different setting)!

@paulgarrett in this case the dual of a Fréchet space is NEVER a Fréchet space, hence that theorem cannot be of use

Yes, that is actually the definition used by Bastiani, Hamilton etc; I'm interested in what we lose by asking this "weak" continuity. For example, if the dual topology is finer than the weak* one (that is, always), we have continuity separately in the two arguments, which suffices for a number of proofs (e.g. the mean value theorem)