Sorry, it is likely that you'll have to reformat. I apologize for telling you that the way to do what you want is to do what you want to avoid doing.

The link to previous/same as previous setting refers to the equivalent header/footer in the previous Section of the document, not to a different header/footer in the same Section.

You could try selecting the text in your first-page header and bookmarking the selection. Then insert a Cross-Reference in the principal header. I expect you will have the same problem, though.

Take a look at my Header/Footer Settings Recap. The only way to get the same content in both parts is to copy the 1st-page header in some way, there are two separate document parts involved. Their contents are both in the same Section formatting, but they are not otherwise linked.

If your formatting were done using Styles, this would probably be a piece of cake.
Importance of Styles in Word


Aside: It is surprising to me that you can have a reference to a bookmark in an inactive header, but I tested it before posting this.