if I have not caused damages to the tenant, could there be any legal repercussions?
If by "arrang[ing] an inspection" you mean that landlord and tenant agreed to have the premature inspection, then the tenant has no cause of action against the landlord. I am not knowledgeable of Australia legislation, but I highly doubt that the statute prohibits the parties to willfully agree on having that premature inspection.
Note that if the tenant received the notice and did not object to it, then that would constitute tenant's implied agreement to have the extra inspection.
Could the tenant claim the costs of cleaning the mould as "damages" for my breach of contract under the "but for" test?
No. The tenant's duty to remove the mould applies constantly (more precisely: with reasonable or practicable frequency) and irrespective of when the inspections are conducted.
Likewise, it is reasonable to enforce the tenant's duty from the standpoint that the effects of mould (such as a reduced value of the property) materialize regardless of inspections schedule.