Would that count as privileged information, such that the lawyer had inadvertently waived privilege?
The attorney-client privilege is something that only the client can waive. The only way the lawyer's "mistaken" tweet would forfeit the privilege is a scenario where the client instructed or consented to the lawyer's publication. The client's consent or instruction for that is a form of waiver of the privilege.
My understanding is that privilege only applies to legal advice, and thus wouldn't cover PR advise like that
Actual or staged mistakes aside, the lawyer's suggestion to her client can certainly qualify as legal advice insofar as it might be in preparation for potential litigation or ongoing litigation. Therefore, suggestions of that type would be protected by the privilege.