pull non-wp site into content area of wordpress -or- pull wp header into non wp site?

Tight integration (loading either inside of other one) without much functionality overlap might come with considerable performance penalty. It might or might not be relevant for you.

Since the issue can be abstracted to “how to transfer fragment of page to external site” I would probably generate that fragment in WP and make other side retrieve in some way. For example via Ajax request or just reading file with cached fragment.

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