Welcome to FT,
bjarmstr!
I hope we aren't scaring you off with all the details and "yes, buts..." as you plan your family's great adventure.
So having said that, I'll throw in more "yes, buts..."
You're probably aware by now that RTW tickets carry different prices depending on where you start and end the trip. Sadly, Canada is among the more expensive origination countries for all classes of Oneworld RTW products - Oneworld Explorer, Global Explorer, etc., in economy, business and first class categories.
One thing to consider might be the potential cost-savings you could achieve by starting your trip somewhere else. For example, at today's prices and exchange rates, a 34,000-mile Global Explorer ticket in economy bought in Canada is C$5999 plus taxes and fees. A 4-continent Oneworld Explorer is C$5099++.
The same tickets bought in Spain, using todays C$/€ exchange would cost C$4360 and C$3613 respectively, i.e. around $1600 cheaper per person.
Obviously you'd have to get to Spain to start the trip, but I can't imagine a situation where you couldn't manage that for under $1600 per person.
If you went with the cheaper (Oneworld Explorer) option and avoided North America entirely on your RTW, you could do it all within the boundaries of a 4-continent ticket. A sample route could be, e.g., MAD-EZE-LIM-IPC-SCL-AKL-SYD-MNL-HKG-SGN-HKG-BKK-AMM-CAI-MAD. That would add Buenos Aires and still give you two more segments to use up somewhere (e.g. within Australia or South America) and it wouldn't violate the rule regarding only two stopovers in the continent of origin (Europe in this case) as you'd be starting and ending in Spain, with AMM and CAI as your "European" stopovers.
Using the Oneworld Explorer for maximum savings and more mileage-earning options, you could, for example, substitute the Queensland coast/Great Barrier Reef area for PPT, or possibly Lord Howe Island, in order to put some "South Pacific" back into the mix.
Regarding the limitations on "L" fares, this is highly variable from one carrier to another, and on some routes. I agree that getting four "L" seats on some flights, e.g. SCL-AKL, can be very tough, but not impossible. If, however, your budget allowed, you could also consider buying 2 business class tickets and 2 economy tickets; the miles earned by the people flying up front could easily "pay" for the upgrade differential in future award travel, not to mention the cost of the Calgary - Europe positioning flights. And if your family is anything like mine, with a year of traveling together a little "adult" time in the quiet, comfy front cabins once in a while might not be such a bad thing.