You will have a much easier time finding comfortable, spacious lodging within your budget offsite. Even in value season with some of the discounts available, two rooms at All Stars or Pop Century plus 12% taxes will end up more than $150. Even if you snag an "upsell" discount on a Cabin At Fort Wilderness (with just one bathroom for the whole family!

) plus 11% taxes, you will still pay more than $150. Those are your least expensive options onsite. If you really wish to stay onsite, you'll have no other choice but to pay more. If you expect to split up the family at times, with some returning to the resort while others continue to tour the parks, staying onsite may be worth the added expense to you.
Personally, I'd go with a lower priced option very nearby, such as the HIFS mentioned. A Kidsuite sleeps 6-7 (bunk room w/kid-trundle, MBR and LR w/sleeper-sofa and mini-kitchenette but only one bathroom). Buena Vista Suites has 2-BR suites that would work too. Both of these places offer a full breakfast for free each morning and both have shuttle services ($) though I don't know how often.
Sheraton Vistana, Cypress Pointe and any of the Marriott "Palms" Villas located at Marriott Orlando World Center are all timeshare resorts located very close to Disney property without a traffic problem. All offer at least 2-BR/2-bath villas that will accommodate your family size VERY comfortably. You may find a rental week from an owner that would fit easily within your budget and a little room to spare.
All of these resorts and hotels are located close enough that if you ever need to split up, the group left behind could take a Disney bus to Downtown Disney (or Epcot), then take a pretty inexpensive taxi cab ride back to your resort or hotel. It's only about 2-3 miles (roads too busy to walk). Hope that helps. Feel free to email me for more specific suggestions.