Then figure out how to bring the cart context to the product page in order to be able to say "Shipping will be +$2 if you add this to cart"

I imagine this could slow down the page noticeably, if you have to hit the shipper's api for every product.

EDIT: sibling comment pointed out, the origin warehouse may be unknown until you have the full order. so now you're figuring out which warehouse you'd hypothetically ship from if the customer bought each product on the page.