Can I send a bookmark with a book using Media Mail?

Yes. Postal regulations allow for “incidental first-class mail” to be included if it “is closely associated with but secondary to the host piece.” This includes a bill for the product, an account statement, or a personal message or greeting. While bookmarks are not specifically mentioned, a slip of paper should be considered incidental.
The complete regulation can be found in the USPS’ Domestic Mail Manual: Section 270 Commercial Mail, Media Mail, and Library Mail; subsection 273 Prices and Eligibility; 6.2 Incidental First-Class Mail Attachments and Enclosures (https://pe.usps.com/text/dmm300/273.htm):

“Incidental First-Class Mail matter may be enclosed in or attached to any Media Mail or any Library Mail piece without payment of First-Class Mail postage. An incidental First-Class Mail attachment or enclosure must be matter that, if mailed separately, would require First-Class Mail postage, is closely associated with but secondary to the host piece, and is prepared to not interfere with postal processing. An incidental First-Class Mail attachment or enclosure may be a bill for the product or publication, a statement of account for past products or publications, or a personal message or greeting included with a product, publication, or parcel. Postage at the applicable Media Mail or Library Mail price for the host piece is based on the combined weight of the host piece and the incidental First-Class Mail attachment or enclosure.”

One caveat: The post office frowns on inserting advertising, so there is still some wiggle room for an officious clerk to be a pain about this.

The best practice you can use is to reduce the chance of an inspection. If you’re sending one book, package it, not in a box, but a padded mailer. Tuck the bookmark in so it cannot be easily seen (if possible). If the handler feels the package and only feels the book, this reduces the chance of it being open (a box with stuff rattling inside, on the other hand …).

But I feel this is a lot of worry about nothing. I have used Media Mail for years, both for shipping books from Peschel Press and when I was running auctions on eBay, and never had a package rejected (or inspected, so far as I know). If the post office rejects a package because of a bookmark or a catalog, I’m not going to worry about it. The worst they can do is not mail it. I’m too old to worry about this going on my Permanent Record.