What is the difference between Asperger Syndrome and ASD Level 1?

No. It was once the dividing line between 'high functioning autism', with speech delay, and Asperger's, without speech delay. If they are using DSM-5 ASD, they really need to read the diagnostic criteria contained in it and not just invent stuff. No-one now gives diagnoses of Asperger's, it is all ASD (or ASC). The criteria they are quoting do not exist for diagnosis of ASD Level 1. Diagnostic criteria are summarised as: “persistent difficulties with social communication and social interaction” and “restricted and repetitive patterns of behaviours, activities or interests” (this includes sensory behaviour), present since early childhood, to the extent that these “limit and impair everyday functioning”.  Nowhere does this include delay in speech and language acquisition or low academic achievement. At best the school are putting their own limits on to whom they give support and passing this off as being clinically sanctioned. At worst they are entirely uninformed about ASD diagnostic criteria. Presumably, there is  no-one at the school who is qualified to give a clinical opinion? In which case their criteria are invalid anyway.

A clinical assessment would be the best thing for the child. In the meantime there are a number of AQ tests for autistic traits available online, getting scores within the autistic range might give some leverage over the school.

Martin said:

No-one now gives diagnoses of Asperger's, it is all ASD (or ASC).

Wrong. Diagnoses of AS still take place as ICD-10 is still a current publication.

 “persistent difficulties with social communication" - could be interpreted as speech and language delays.

“limit and impair everyday functioning” - could be interpreted as behind academically in the main subjects.

Kindly expressed! Going from the the following quote I thought otherwise: "‘Asperger’s disorder’, or ‘Asperger’s syndrome’, is actually no longer an official diagnosis in the UK (or the USA, for that matter). Since 2013 this was dropped in favour of ‘autism spectrum disorder’ (ASD)."

“limit and impair everyday functioning” - could equally be interpreted as having an aversion to noisy places or crowds, or an inability to follow conversations when levels of background noise are high. I have these aversions and inabilities, and many others, that NT people do not usually have, and yes they limit and impair my everyday functioning.

To be blunt, as you seem to use bluntness yourself, without a clinical diagnosis you haven't a leg to stand on in regard to getting accommodations from the school. In the absence of a diagnosis they are not legally obliged to put anything in place, and you are then reliant on their sympathy and helpfulness, which appears to be non-existent. 

Kindly expressed! Going from the the following quote I thought otherwise: "‘Asperger’s disorder’, or ‘Asperger’s syndrome’, is actually no longer an official diagnosis in the UK (or the USA, for that matter). Since 2013 this was dropped in favour of ‘autism spectrum disorder’ (ASD)."

“limit and impair everyday functioning” - could equally be interpreted as having an aversion to noisy places or crowds, or an inability to follow conversations when levels of background noise are high. I have these aversions and inabilities, and many others, that NT people do not usually have, and yes they limit and impair my everyday functioning.

To be blunt, as you seem to use bluntness yourself, without a clinical diagnosis you haven't a leg to stand on in regard to getting accommodations from the school. In the absence of a diagnosis they are not legally obliged to put anything in place, and you are then reliant on their sympathy and helpfulness, which appears to be non-existent.