Dear win128,
Alleged breach of agreement - Unfortunately no - I read each document - there is no law broken by the insurer per se. I am not even convinced notice is required for the subject matter. And you don't really have evidence in the attachments above - that letter from the insurer isn't part of the agreement you had with them, or arguably isn't inconsistent with the terms and conditions/definition of appropriate ground to going overseas - and to be more thorough I even checked HBF's terms and conditions between 2010-now online - I didn't identify anything that suggests the insurer must agree with your request to extend the suspension of insurance without penalty from 1 year to 4 years.
Alleged unfair dealing with you - unclear - maybe you can prove in a tribunal that the insurer's decision, while seems to be legal, would create hardship/unfair in your case/improper/didn't account for all details. Need much more details.
Also - what is this "case description" document? Are you seeking proper legal advice? If so, you need to provide more details - e.g. date of insurance commencement, documents you had signed, copy of all or any notice received from the insurer, how and what was done when you were dealing internally with the insurer. You were vague in the documents above, but it seems relevant if you have any evidence why you are overseas and want to extend and extra 3 years, and why even with 1+ years overseas that you are travelling and not living overseas (e.g. ATO definition is >1/2 year/183 days outside of Australia to be a non-tax resident, insurer likely would bring this up saying they are generous already etc).
I am not suggesting you have no case - I am suggesting if you wish to pursue further e.g. Commonwealth Ombudsman complaint, you need to focus a lot more on evidence that the insurer erred in considering you to have resided permanently overseas, or the insurer erred in how they had dealt with your dispute. Right now, with the information you put to us, you would be disadvantaged before a tribunal.
FYI - you have 3 months from the date of the final response from the insurer if you wish to bring complaint, that presumably means 12 November 2022.
Good luck!
-Nighthelyn
(Edit: AFCA doesn't deal with private health insurance - it is actually Commonwealth Ombudsman Private Health Division that provides external dispute resolution complaints. I've updated my answers).