Yeah nah dont fight with the new company. Fight with the old company, just tell them you'll work an extra 4 days to complete handover tasks and shit. Dont ask, just tell them.
But then since you got a massive salary hike at new company (according to hike thread), this is still going to cost you 4 days at difference between two salaries so you are still up shit creek, not much you can do about it, just keep whinging on geo and take comfort in knowing that your life experiences do provide some entertaining reading for us here