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I answer myself, just in case someone finds this and have the same problem:

My bad: the solution to the "strange" fundamental weights is explained in

Following the example:

sage: [fw.coerce_to_sl() for fw in A2.fundamental_weights()]
[(2/3, -1/3, -1/3), (1/3, 1/3, -2/3)]

Anyway, I am considering proposing to add a style='coroots' option to weight_multiplicities() to get the output in 'Dynkin labels'.