This is unusual for these boards, because emotional issues are relatively rare compared with eg ASD
If you are looking for a maintained mainstream then you are entitled to your choice unless, among other things, it is incompatible with the efficient education of others
IPSEA have the following advice about schools that claim to be full-
incompatibility with the education of others This is the test which is often referred to when a school is said to be “full”. However what the test requires, if it is to be used successfully by an LA, is that it is likely that adding just one more child to this school or other institution will create incompatibility with the education of others. This is not the same as showing that the addition of a child or young person will in some way impact upon the education of others. The test is whether the impact is so great that the child or young person attending the school or other institution would result in an incompatibility which cannot be resolved by the special educational provision being put in place as detailed in the EHC plan. The facts need to be examined in each case and this has been found by the courts to be a high bar.
So you could try to persuade a tribunal that you should have your parental choice.
I am concerned by your existing school refusing to do more. Do you think that your present school could achieve more if it had more finance? It may be that they are out of money and need more from the LA to do more interventions, That might imply a reassessment.
But if your other school seems to fit the bill when you go round, have a think about what the differences are that make one seem better. If it is an environment issue, ie the presence of smaller classes or a break out room, then it may be that no amount of extra money would make the present school tick all the boxes of things that dc needs.
Also 'out of catchment' is not relevant, although going to a different LA can turn up some additional issues.
Hope the school visit goes well
Good Luck