The driver that Battlefront is talking about is the HD 4400 driver. xx.xx.14.4264 is a HD 4400 driver number, not HD 4000. The 4276 driver you have installed is the latest driver for your graphics set.
Battlefront doesn't work on the HD 4000. I've already reported the issue here but I doubt it will be fixed. Star Wars Battlefront Intel HD 4000 crashes with E_INVALIDARG error
We have already provided you the correct answer in regard to this matter, I'm closing this thread as duplicated.
I definitely understand your situation but like mention before if you run the game out the official requirements any issue you might get will be expected and not supported. Maybe you could try contacting these users, the ones you mentioned to get feedback on how to run the game out the official requirements.
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