If you read the page you linked to, you can see that the antigen is immobilised. Antibodies from the patient serum bind to this antigen, and are in turn detected by a subtype specific antibody which is either specific to IgG, IgM or IgA, depending on the kit you are using. They list an IgA-specific test kit - are these no longer available? You might ask the company whether you could just get their IgA-specific second antibody to use with your existing IgG kit, but depending on the exact details of the assay, they might not warrant the accuracy of this combo. You could theoretically use a labelled anti-human IgA antibody from another source for detection, but you would need to optimise and verify the resulting new system from scratch.

Thank you very much for the information. Now I understand the problem. For the Omicron variant only IgG was available. Your suggestion to ask for the IgA specific second antibody sounds like a potential solution. Thank you.

You can try to find a secondary antibody that is suitable for detecting IgA and replace the secondary antibody in the original IgG kit.