Short answer - Yes, the lawyer can use that term.
What he's not allowed to is call himself "a barrister"
as though that's what he is.
Long answer - In NSW, professional titles are prescribed in Rule 9 of the Legal Profession Uniform General Rules 2015 .
The term "Solicitor and Barrister" is provide for in item 2 thereof.
The requirement is to have Practising Certificate. That he has a supervised PC is not relevant.
What he's not allowed to is call himself "a barrister", as though he's a wig-and-gown-clad
full time specialist court advocate.
(And, there are separate rules about solicitors wearing robes etc)
The terms "Solicitor and Barrister" and "Barrister and Solicitor" are both OK, and interchangeable.
Personally, in day to day life, I'm a solicitor, even though I get on my feet quite often.
For myself, I only use the term "Solicitor and Barrister" when working overseas (such as when I work in China).