Cost and legalities
Cost of another plebiscite to change the tune (with new words) prohibitive
Following precedent and convention, if the tune of the National Anthem was to be changed, another plebiscite would be required.
This would be prohibitively expensive for the taxpayer.
The present government’s earlier plan in 2017 for a national plebiscite on same-sex marriage provides a guide as to the expected cost of a national plebiscite, if one was to be conducted to change the tune of the National Anthem.
The cost of a voluntary postal plebiscite on the same sex marriage issue was estimated to cost taxpayers $122 million.
The cost of a full plebiscite on the same sex marriage issue was estimated to be much higher. Consultants PricewaterhouseCoopers (PwC) estimated the cost of a full plebiscite to be $525 million, made up of $160 million for the ballot itself, $66 million to fund the “yes” and “no” cases, and $281 million in lost productivity  (with $18 million presumably for contingencies).
In the case of the National Anthem, if a change to the tune was to be considered, there would also be very significant additional costs associated with the vetting of public submissions and selection of a new tune or tunes (with accompanying new words), prior to this being put to a national ballot conducted as a plebiscite.
A plebiscite not necessary to merely change the words
However, following precedent and convention a plebiscite would not be necessary to merely change the words of the present National Anthem.
All of the very large costs of conducting a plebiscite would be unnecessary.
As to the proposed new RAP lyrics, or something like them, there is no reason why, perhaps with modification, the process should not follow the path set by the precedent and convention adopted by the Fraser and Hawke governments.
Suggested modifications may include for example, any appropriate consultation on the new words considered necessary at the time of introduction and possibly additional support provided by a vote in Federal Parliament as a preliminary step to the making of a proclamation by the Governor-General to adopt the new lyrics, pursuant to the exercise of executive power under section 61 of the Australian Constitution.
Exercise of executive power
In relation to use of the executive power under section 61 of the Australian Constitution to introduce new lyrics for the National Anthem, Brennan J of the High Court reinforced this approach in Davis v The Commonwealth (1988) 166 CLR 79 at  where he said: “The end and purpose of the Constitution is to sustain the nation. If the executive power of the Commonwealth extends to the protection of the nation against forces which would weaken it, it extends to the advancement of the nation whereby its strength is fostered. There is no reason to restrict the executive power of the Commonwealth to matters within the heads of legislative power. So cramped a construction of the power would deny to the Australian people many of the symbols of nationhood - a flag or anthem, for example - or the benefit of many national initiatives in science, literature and the arts.” See Too Victoria v Commonwealth (1975) 134 CLR 338 at 397; R v Duncan (1983) 158 CLR 535 at 560; R v Hughes (2000) 202 CLR 535 at [554-555]; and Pape v Federal Commissioner of Taxation (2009) 238 CLR 1.
 Malcolm Farr news.com.au (last observed, 15 August 2017).