This is probably a matter of two different things that operate in unusual ways not accounting for one another.
Usually, when a construct is reduced below 0 HP, it is destroyed. The limitation on rapid repair covers this case, and doesn’t work.
But Clockwork Automaton has the creature become Unconscious, rather than destroyed, when below 0 HP. The whole point of the ability is so that they can be repaired more easily than typical resurrections; it explicitly says they “can be restored by use of the make whole spell [...] or other repair spells for constructs.” That said, I note that make whole does not use the same line as rapid repair about not working for constructs that are destroyed or at 0 HP.
Ultimately, the fact that rapid repair does not work for constructs at 0 HP (when a construct is normally inert but not destroyed) implies to me that it should not work for Clockwork Automatons below 0 HP. RAW, however, this unusual exception to the typical construct rules is not explicitly covered by rapid repair.
But this is a specific limitation of rapid repair; other spells that lack the line about not working on constructs with 0 HP, I would allow to work, per Inorganic Functioning’s statement that Clockwork Automatons “can be restored by [...] other repair spells for constructs.”
Regarding extra costs, that much, at least, is clear: “Regardless of the method used,” repairing a Clockwork Automaton with 0 or less HP is going to be expensive. Even if the method used would usually be free.