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Is there some straight-forward way to classify dynamic obstacle from the cost map, so I can send zero_velocity from DWA upon detecting one?

As I said in the question you link to: No (not to my knowledge, at least).

Or do I have to write a custom local planner?

You could try the teb_local_planner (following Tut 3 on this page) and check if this helps.

If only following the global plan without obstacle avoidance, only waiting, is your only goal, I'd go for a custom planner. This could have a simple PD controller for path following and a little look-ahead stopping movement once it encounters an obstacle...