Short answer is no; you're looking for a unicorn.
1) The biggest reason is there are very few hotels that offer reasonable shuttles to the parks. If you stay onsite or at one of the DS hotels, you'll get generally reliable shuttles. If you stay farther afield, the hotels that offer "free park shuttles" actually mean they'll have one or two a day, not necessarily at convenient times, and will often only go to one park. So the shuttle drops you at the Epcot parking lot, and you're expected to transfer to Disney transportation to get to the park you want to visit. While a few places do offer better shuttle service (like the Hilton Bonnet Creek), they are above your stated budget.
2) Party size might be an issue. We are also a family of 5, and have found that many hotels expect more than 4 people to get two rooms or a suite. There are some exceptions that allow for 5, but you'll have to look around for them.
3) There are lots of inexpensive hotels in the Disney area. Many of the cheaper ones have very mixed reviews. Your later post indicates you'd like a nice pool, and for it to be a generally nice resort, not just a crash pad in close proximity to WDW. Cheap hotels really close to Disney are likely to be sketchy. If you go far enough away from the parks so that you get a nice hotel for a reasonable price, the hotel probably won't have a shuttle.
That said, it might be possible to find a deal that checks your boxes, but it will probably be more luck than anything. Hotels change prices all the time, so you might find a deal, depending on your dates and if you hit it just right. If you look on the thread for Priceline bids and results, you'll see people do find some amazing deals, but they tend to be one-offs rather than a particular hotel you can just call up and get everything on your list.
My two cents: Either raise your price cap a bit to get a nice Disney Springs hotel that offers good shuttle service, or book a nice hotel a bit further afield, and factor in an extra $22/day for parking at the parks. With the high price of Disney park tickets, I can't imagine wasting literally hours of my park day in unnecessary transit, which is what you'll do if you try to make a cheap hotel's shuttle work for you. For $22 you could park at any of the Theme Parks, and be there in minutes. Go to a site like Priceline, and do a search for your dates. If you find something with decent reviews for $50/night and free parking, then that's only $72/night with Disney Parking.