I haven't figured out the quoting in the new software yet, so OP, you are purple....
DS1 is in 1st grade and has no real intrest in doing anything but playing and watching tv. When I try to get him to "do school" it is a huge fight and he can't focus for more than 5 min at a time.
When our DS1 was in first grade (last year) I had similar problems. The way I addressed it was to lock up the remote. No TV until school work was done. I helped him through it five minutes at a time, with frequent short breaks, but I made him keep coming back until it was done.
I don't feel like I push much. We only really do actual work maybe an average of an hour or 2 a day depending on how long he fights me.
However, when I started this with him, I was only having him complete 30 minutes of actual work time, in five minute chunks. Two hours would not have been mentally possible for him then. As his stamina increased and he could work longer chunks, I increased the total work time. But even now I don't think he does two full hours of work, except on the days he has a private lesson or tutor in addition to what he does at home. He has become pretty efficient at getting his work done now, and he doesn't need two hours to do it all.
We are going through a charter school so I push somethings that are not a big deal to me to make the school happy, but not being in the charter is not an option for us for several reasons.
I can't address why you believe you can't do it without the charter, but I will say that a charter would absolutely not have worked for grade 1 or 2 for our son. It would have pushed him too much in areas that were too much of a struggle for him, leading to frustration and breakdown. It has been necessary for him to be able to go at his own pace and get all the help he needs in basic skills. Yes this makes him a bit behind now, but I don't believe there is an alternative - it's not possible to rush real learning. We will work on the basics until they are no longer a struggle, and then he can catch up. I think a charter would have been terrible for him. I don't know what your reasons are for feeling you have to be in a charter, but I believe they can make homeschooling more difficult than it needs to be.
He is not interested in doing anything the resembles school. He has told me he wants to be adopted by a family that won't make him do any school.
If my child was flat refusing to do any school work at all, and our entire day was a huge negative power struggle, then I would keep school in my head as a last resort but a real option. With our son, I was able to remedy this by adding more structure to our day. I required school work to be done immediately after having breakfast and getting dressed, no TV, no playtime, no other options, just five minutes of work alternated with breaks of hopping or running around, until it was all done. I still use this structure. DS2 plays by himself in the basement, sometimes with a video, until DS1 is finished. He has the option to be with us and do his own quiet activities but would rather go play alone. Often he interrupts, and I get him back to his playing and we go on...DS1 still has to finish his work before he's off the hook for it. Remember I am not talking about hours here - just long enough to do a bit in each area, often not even one hour of total time, and this is grade 2. For grade 1 I would make it shorter.
I just don't know what to do. I feel so inadequate to teach him, deal with the 4 year old and 5 month old. Next year it is just going to be worse with a 2nd grader, Kindy, and a 1 yr old to chase. I feel like giving up but am scared to put him in public school. Plus I'd still have to fight him on homework.
I think these can be difficult ages - mine are 7.5 and 5, and I do feel younger siblings make it more difficult. Sometimes DS1 tries to use DS2's distraction as an excuse to not do his work. I am not doing a much as I would like with DS2 because DS1 still needs so much help. I am hoping that by next year DS1 will be able to do more work independently, even just for 20 minutes or so, and I can do more focused work with DS2. DS2 is taking a once per week one hour phonics class through our park district so that takes some pressure off me. I think it's just tough with two active young kids who would rather play. Imposing some structure for the first couple hours of the day, and being realistic about how long one can focus, and being relaxed about how quickly they progress, has been the combination that has worked for us. I tell myself that two years from now, this will all look different, and that getting through this stage with these particular ages and getting off the ground with academics may be the most difficult part of homeschooling I go through.