I don't think you can do this through the API. The company search facility is more based on finding a specific existing company. You might want to look into the XML Gateway NameSearch query, as you can use this to return not just exact matches, but also what Companies House considers "same as" an existing name.
I'm assuming this is the service consumed by the CH name checker (https://find-and-update.company-information.service.gov.uk/company-name-availability) based on the results this search gives and the fact that I don't think you could currently extract this information using the API.
Links that may help:
XML Gateway: https://xmlgw.companieshouse.gov.uk/
Schemas (including NameSearch): https://xmlgw.companieshouse.gov.uk/v1-0/xmlgw/SchemaStatusOutput