But Jake is lovely! - I actually prefer it to Jacob itself but would put Jacob on the birth certificate. If you call him Jacob all the time, probably family at least will do the same, and not shorten it. However when he goes to school, you kind of lose control over it a bit - does it matter?
Actually knew one that was a Kuba (pronounce koo-ba, not like the island of Cuba) but he was from a Polish family so it was probably actually Jakub.
It’s up to you and then the child if they want their name shortened. If you like Jacob tell people that’s their name and correct them if they shorten it. If you don’t think you could do that then choose a different name.