Just say "we pay the local market rate where you live" and if that's true, you won't have any problem with your offers. If the developer in Canberra expects a Mountain View salary they will presumably explain to you why that's in your interest, and you can be convinced or not. Everyone else will be OK with the local market rate, because markets.
Publishing your "Canberra Disadvantage Index" doesn't seem likely to get you anything but people complaining on the Internet.
(Though I do wonder what GitLab would do if they find someone in Canberra they really want to hire and she already makes a San Francisco wage. My guess is they hire her and ask her to keep quiet about the salary.)
For starters, you'd have to be okay with living a lie to your coworkers and putting your career at risk if you ever let the facade slip. You can't ever talk about local events at the virtual water cooler. Do you live in the same time zone as one of the high-cost-of-living cities? If not, you can't ever mention what time your kids go to school either. Is your employer ever going to fly you out to conferences or all-hands meetings? They're either buying your plane ticket directly or expecting to see a receipt; good luck explaining why you never fly out of your "home" airport. Does your employer report your earnings to the IRS? If so, you'll likely be committing some sort of tax evasion or at least having to submit a ton of corrective paperwork each year. Do they provide insurance? You'll have to explain to the provider why your address isn't what your company thinks it is or else risk voiding your coverage and/or committing insurance fraud. And so on, and so forth.
All in all, it's probably doable, but very few people would consider it worth the effort unless they are pathologically inclined towards fraud in the first place.
Bear in mind though that, for example, in the UK there's a differential of 50% for living 1-2 hours' train ride out.
I don't even know if towns I've previously lived in and commuted into London from would be counted as "London" by their calculator.
Only one of those two things correlates strongly with market wages, though. But you may be right that it may be more transparent to just say "cost of labor" rather than "cost of living".