Well, the GC system works differently that what you're picturing. The backcountry management strategy, to minimize impact, is based on limiting it to small groups of hikers and limiting the number of these groups. A permit is issued to the group leader and that's the only name on the permit, and the permit is not valid without the permit holder. The permit lists each camp or use area in the itinerary and doesn't allow for deviation.
So, while it may seem like it would be more "efficient" to be able to mix-and-match to maximize the campsite limit, it's really not the site that has the limit, it's the group size, and only so many groups are allowed in one camp/use area on any given night. The management plan doesn't expect or want to maximize the group size. If every permit was for 6 people, the number of people in the use area would far exceed what it could absorb, in terms of impact.
I hope that made some sense.
That said, if you are able to find permitted hikers who are willing to share their site with you, I'm guessing that *most* rangers would let you stay, even if they weren't happy about it. But, personally, as a parent hiking with a child, it's not something I'd want to count on, especially in summer hiking conditions. I hope you're aware that anytime between mid-May and mid-Sept is potential for very hot conditions below the rim, with temperatures above 100. Be prepared!
Btw, there is no way to "register" two extra people for just one night on a permit, so you would have to pay for two extra people for the entire itinerary. In other words, if you found 3 groups with 4-night itineraries, each willing to add you to their permit and share one night's camp with you, it will cost you $40 for *each* night.