Star Wars Battlefront 3 works on Intel HD 4000 for me. Here is how:
I have a Lenovo U310 with an Intel i5, 8 GB RAM and integrated Intel HD 4000, running Windows 7 64 bit. Starting Battlefront either from within Origin or from a desktop icon never went past the driver notification window shown in your message when modifying the usual Windows compatibility options. In desperation I then did several things in accordance with different online posts:
* I activated speech recognition
* I downgraded my up-to-date Intel graphics driver to the older 10.18.10.3277 version (http://support.lenovo.com/cy/en/downloads/ds038195), which is not specifically indicated for my laptop.
* I added the "-novid -noborder -windowed" game options in Origin
* I made sure no secondary screen was plugged into my laptop's HDMI port.
In line with the driver installation I had to restart my laptop.
It worked. Starting SWBF3 from within Origin I still get an obsolete-driver message, BUT the game starts. Choosing from in-game Settings the native screen resolution of my laptop (1366x768) and between 50% to 70% rendering resolution gives playable frame rates. The game looks terrible this way, but this is an interim solution until my son can play his game on a newly ordered desktop machine.
I now have to remove some of the above options to find out really what was the impediment to the game starting. That would be me working backwards to dissect the problem. For anyone wanting to move forward, downgrading to the Intel driver version linked above might be a good starting point.
Good luck!
Subsequent edit:
Deactivating speech recognition in Windows still allows the program to run
Removing game options in Origin does not seem to interfere either