There are a couple of ways that schools deal with this issue. Many inclusive schools, in their effort to keep kids in the regular classroom as much as possible, try to provide skill instruction (in reading and writing, for example) in the regular class as part of the ongoing instruction. This kind of "integrated" teaching can work if students' basic skills are not too weak and if teachers are well-supported. If this model doesn't provide sufficient support, then students need instruction apart from the regular class. The problem is, as you have discovered, that this extra help needs to fit in somewhere. Some elementary schools or middle schools (where credit requirements are not such a big issue) make it possible for kids to have intensive support in small groups or individually in place of special subjects like music or art. Of course there is a "cost" here, in that children don't get to experience this very important part of the day. As a solution, some parents try to build the arts back into a child's life by getting her involved in after-school or weekend activities. Some schools build in a period a day that can be used for enrichment activities or support, depending on a child's needs. Other schools provide special support during a part of the child's lunch period, but these are usually so brief that this is not a great option.
Very few schools have special supports available in the morning before academic classes, but some do have an "X" period after school to offer such supports. You could push for it, but the school is not obligated to provide it. In some cases this model can't work because of kids' after-school schedules, restrictions imposed by teacher contracts or unions, or transportation constraints. While after-school support can be helpful, it can come at a time when a child is overworked, tired, or stressed. If private tutoring hasn't been successful because it was not connected with what was going on in school, then try to build this into the IEP. Perhaps if you use a tutor who has worked with the school before, the bridge between school and tutor will be stronger and the instruction will be better coordinated.