Julia - You still haven't understood how this all works, as a result of which you are completely misinterpreting what the school has told you.
The LA still receives funding from the government for ed psych, SEN assessments, statementing, monitoring SEN provision, PRUs, etc. The LA has to provide these services free of charge to all the schools in the area including academies. So this school does have access to these services regardless of whether or not it uses them.
The academy does not receive any money for ed psych, SEN assessments, etc. because those things are provided by the LA. If it had to provide those services itself it would receive more money. Instead, its contribution to these central services is taken away by central government and given directly to the LA.
To illustrate with some figures (which will be wildly inaccurate but give the general idea), imagine that the government pays grants totalling £7000 per pupil to your LA and that the LA deducts 10% of that for central services of all kinds and passes on £6300 to the schools. Of the £700 deducted by the LA, let us say that £200 is for ed psych, SEN assessments, statementing, etc.
When a school converts to academy status it will receive its funding direct from central government. But that doesn't mean the school will be getting £7000 per pupil. It will get £6800 per pupil at most, with the LA still receiving £200 for each pupil at the academy for ed psych, etc. In fact the academy will get less than £6800 per pupil as the LA will still receive funding for certain other central services that remain their responsibility such as co-ordinated admissions.
The academy is effectively paying £200 per pupil to the LA for SEN services. It isn't paying that money directly but that is how much its grant would go up if the LA was not providing SEN services to the academy.
The important point is that the LA has precisely the same amount to spend on these aspects of SEN services as it would have had if the school had not converted to an academy.
The only thing the LA may be providing which the school will not have access to once it becomes an academy is SEN support services (I say "may be providing" as many LAs don't do this, leaving it up to individual schools to do it themselves). As it has to provide its own support services the academy will receive some additional funding based on the number of pupils it has with SEN. If there are 100 pupils with SEN in the LA and only one of these pupils goes to this school, the school will receive 1% of the funding that would otherwise have gone to the LA for SEN support services. The amount of money available per pupil for SEN support services in the other schools is therefore exactly the same as it would have been if the school was not an academy.
Other schools are not losing out on SEN services in any way as a result of this school becoming an academy.