I don't quite agree. Beatrice and Eugenie got HRHs because they are granddaughters in paternal line of a British Sovereign and all children of a Sovereign's son are HRHs in the UK. It has nothing to do with being a "working royal" and it is a rule that has been in place for a long time, actuallv before 1917, since the Hanoverian period. Andrew didn't have to fight for it as there are no indications AFAIK that the Queen planned to change that rule during her reign.
Similarly, Edward's children, who are in the same position that Archie will be when Charles is King and in the same position as Beatrice and Eugenie now, could (actually should) be HRHs too. But, in their case, conversely, their parents "fought", or rather asked the Queen (which is a more accurate description) to have them styled only as children of a peer, which she agreed to without, however, making any formal changes to current rules by issuing new LPs.
I interpret the way James and Louise's case was handled as a signal that the Queen saw it as an exception that applied to Edward's children only by special dispensation of her will and pleasure, and not as a rule of general application, such as e.g. all children of the eldest living son of the Prince of Wales becoming HRHs, which was effected by means of new LPs in 2012.