Always appeal parts 2 3 and 4 (or EHCP equivalent letters).
You can make an appeal smaller later, but it is much harder to make it bigger!
If your chosen school is independent then it CAN refuse to let your child in and the Tribunal cannot over rule it. But ask where dc is on the waiting list as things are still settling down, and getting a feel for how likely a place is to pop up is important.
If it is maintained, then yes, the Tribunal can insert an extra child in. Even if that means going over the statutory maximum of 30 in a key satge 1 class (probably irrelevant in your case).
So now it is time to find out what the class sizes are for all the classes through the chosen school, and also what their maximum number of pupils in the school is, and how far under or over that figure they are.
Having done some groundwork, you may want to have a meeting with them to find out whether they would contest an appeal on the argument that 'one more squeezed in cannot make so much difference'. Obviously if your dc's statement comes with a full time one to one added in, then they may actually be quite happy to have an extra pair of hands around. OR they may say that the physical size of the class and the distraction of not one but two extra people would affect the learning of the others.
The other big job is to find alternative schools. There is a real risk that a popular school will convince a Tribunal that they really cannot take another kid, and you don't want to be left with nothing but the LA's choice as the only thing in front of the Tribunal. So I am afraid a good deal of far-and-wide hunting may be needed.
If your new-chosen alternative is independent and vastly more expensive than your old-chosen school, then you may find a place suddenly opens up in your old-chosen 'full' school. It happens.
You will also want evidence to show why the LAs option is not able to meet dc's needs.
With luck, it will be things about the school environment that they simply cannot change eg classes too big and noisy etc.
Otherwise it will be that one lot of staff are just better geared up to do the job than the other. You can ask for a table of qualifications, courses attended and experience etc for both the LA choice and your choice(s). Then there are the OFSTED reports, the SEN policies and a comparison of the other kids in the class and their difficulties (noisy?) - you can ask for this evidence from both schools in general terms, just not personal details.
The frustration really is that a good school will set out all the things that they think dc needs, and then the LA will promptly claim that their school can change to do all of that.
One compelling way to crank up the pressure is to have your own indie expert(s) visit the LA school and pronounce on its inability to meet dc's needs. Ideally they would visit your chosen school(s) and give their views as well. Pricey and no guarantee that it is 100% effective.
Well that is a long old post, but I hope it gets you started.
Don't give up - your gut instinct is likely to be right, which makes fighting for it the right thing to do.
Good Luck