Hi, Kaitlyn! I think your plan is entirely possible, but there are a few things to consider. Let's discuss...
I think the best way for you to get close to your budget is to take advantage of a Special Offer, such as the
Free Disney Dining offer. With this offer, you pay full rate for your Resort hotel room and Park Hopper tickets. You will then receive for free either the Quick-Service Dining Plan, if you choose a Value Resort, or the regular Disney Dining Plan, if you choose a Moderate, Deluxe or Deluxe Villa Resort. You can upgrade the dining plan by paying the difference in price between the plans.
I mention all of this because I ran a mock vacation for the end of August/first of September, and staying at one of the Value Resorts while taking advantage of the Free Dining Special Offer will get you very close to your budget, but just a few dollars over. However, staying at a Value Resort will provide you with the Quick-Service Dining Plan, which is not what you wanted. As I mentioned earlier, you are able to upgrade your dining plan, which would add a little over $400 to the price, bringing the total to just under $3000. A little thing to note here - remember I mentioned that Moderate Resorts get the regular Disney Dining Plan for free with the Free Dining offer? For just a tad bit over $3000, you can stay at a Moderate Resort and receive the dining plan that you want, as well as larger comfy beds and great Resort amenities. I absolutely love Disney's Pop Century Resort. But if I had a chance to stay at Disney's Coronado Springs Resort instead, for around $100 extra, I would scrimp and save wherever possible to do it!
These are just a few thoughts to consider when planning your vacation, Kaitlyn. Please remember that prices vary during different time periods of the year and are subject to change. I hope you are able to put together a magical vacation for your family!
Brandy