Before becoming an assistant headteacher you were subject to teachers'
conditions of service, which include the provision that you have to be available for work under the direction of the head for 1,265 hours per year.
Outside those hours, teachers are at liberty to enter into separate contracts, provided that they do not affect their normal work, for example to do lunchtime supervision or take additional classes.
By entering the leadership group, you accept a change in your conditions of service. You are no longer constrained by hours or days and you are deemed to be a salaried person in respect of all work you undertake on behalf of your employer.
The LEA is probably arguing that you cannot have a separate contract, because what you are doing could be considered as being within your existing one.
One has to say that this all seems very unfair. Some schools have not pushed this too far, allowing, for instance, deputies to continue to draw payment for lunchtime supervision, when, strictly, they are wrong to do so.
I can only suggest that there is absolutely nothing to prevent you giving private tuition at times when it does not affect your normal job.