There are three elements that need to be simultaneously fulfilled:
In the Page Styling, you need to enable both the header & other header — this ensures that ClauseBase copies those elements from the DOCX file. I understand from your question you already have this covered.
In the Page Styling, you also need to make sure that you enable a different header/footer for the first page vs. the rest of the pages. Otherwise ClauseBase will copy the headers from the source DOCX file, but the “other header” will never show actually show up.
Third, you need to make sure that your DOCX file actually has two different headers — otherwise only one of the two headers will get copied from the source .DOCX file.
On my Mac’s MS Word, the highlighted checkbox in the screenshot below must be checked: