Fake the date in header when printing in Chrome?

Yes, by setting your computer's clock backwards. Or by editing the PDF file using one of the dozens of (free and paid) PDF authoring tools that are available.

There are ways to detect that a PDF file has been edited after creation – for example, there are actually a few more timestamps in a PDF file (not just on the page, but also in metadata) – but none of those would reveal whether it was simply "printed" with the clock showing the wrong date.

If there is a need to prove that the PDF was created no later than a specific date, then various "trusted timestamping" services could be used.