It depends on what content is present on your accident form - if there are names and other details from which anothers identity can be derived these should be redacted under GDPR (this is probably where your thought of a written request kicks in a "Subject Access Request").

Personally I would make a note or keep an annotated copy of what is provided.

I note you do not mention who "they" are - is it the i.p., their parent or representative, the union, the local authority or school board.

For a combination form (record of incident and investigation) I would inform the insurers before providing any additional information over and above the record of incident.

A demand of "end of the day" sounds like a barrack room lawyer is poking their nose in waving an idea of blame and claim - direct all communication to the insurer do not engage with such people.

Who had the accident and where?  If at school I would expect the school to write the report and if so they don't need to ask for a copy.

My wife had an accident this year in a hotel and I made sure the hotel completed the accident fom and I received a copy on request.