I'm thinking of buying an extra copy for a friend, with the Black Friday 50% sale. Problem is, the sale is only valid in the US, not EU.
If I buy the US version, would I be able to convert it to the EU?
No you cannot. Each version is locked into a specific region. Please see this answer for more details: How does the StarCraft 2 region lock work?
Actually you can and I did,. Where I live in Canada, games that have a french version must so they sell the EU version (they don't say it on the label).
So I called them, reminded them that Canada was part of North America and if they want to region lock, they could at least sell the proper games in the proper region.
They added a StarCraft 2 North America to my account and I downloaded it.
So yes you can, at least you could.