We are a family of 5 and have avoided all inclusive resorts for that reason. Upgrading to a suite or 2 rooms is cost prohibitive. Here are some options for you:
1. Find a resort you like and call them. Explain your situation and ask for a discount. Not sure this will work, but it is worth a try.
2. Have you considered a cruise instead? They are all-inclusive (except alcohol). My family of 5 is going on an 8 night cruise for under $550 per person (tax and port fees included). When you consider this includes food, it's actually quite an amazing price. Carnival seems to be the most family friendly in terms of offering units for 5 guests. Note, we will be crammed into a very, very tiny room. However, we don't intend to spend much time in the room.
3. If you are going with friends or family who have less than 4 in their party, have them add one of your kids to their reservation. The child doesn't need to sleep in their room. We did this a few years ago at Club Marival in Puerto Vallarta. Our 5th family member was booked with her grandparents and was supposed to stay in their room. We bought an air mattress and asked for extra blankets. Not a problem.
Hope these suggestions help.