With your group it would be so much more convenient to stay on site. You'd be more hands free with not having to drive everywhere. Also, you pay for parking at the parks if you stay off site which I think is $12 a day. That pays for a fridge. You may get a swap for the fridge if you plan ahead enough and that would make it free. And this doesn't count the gasoline for the car going back and forth all the time if you are off site. Finally, the time you lose which is so precious on vacation going back and forth from off-site to the parks is a real killer and deal breaker for me.
If you want to take a nap it's so much easier and better at a Disney resort. Some of the kids nap on the bus even; so, they get more rest for a break. The kids like the theming at Disney better.
The extra magic hours by being an on-site guest is so worth it to stay at Disney. You accomplish so much more by having extra time at the parks.
Try some other avenues for reservations like a
AAA office to check for you on getting your dates. I would go as far as changing my dates a little to get a deal that works. I think if you are a little flexible it will work out for you.
Also, I think it's a better vacation for you if you do get the Disney meal plan. You have the offer of doing quick service this year for a plan which is cheaper and it doesn't require reservations anywhere. The food courts are good for this plan including dinners there. You don't have to pay tip and tax is included in the dining plan. If you go for the better plan then that's even nicer.
I think this is an exceptional year at Disney to take advantage of the deal they are offering. Getting free park tickets for three days will not be matched off site.