So I dug into the INF for this thing. I looked at versions 3131, 3165 and beta version 3214. This is what I found (only listing the 4000 series):
3131 has support for 4000 and 4600
3165 has support for 4000 and 4600
3214 has support for 4000, 4400 and 4600.
So as far as I can tell this is exactly why this driver is not loading. There is no 4400 support in the two production drivers at all. I would have to say this is exactly why it says my 4400 does not meet the requirements for the driver and is exactly why the beta driver installs (in my case, at least tries).
Production INF:
DiskId = "Intel(R) Graphics Media Accelerator Driver for Windows"
Intel = "Intel Corporation"
iIVBGD0 = "Intel(R) HD Graphics 4000"
iIVBGM0 = "Intel(R) HD Graphics 4000"
iIVBGD0SRV = "Intel(R) HD Graphics"
iIVBGD0GT1 = "Intel(R) HD Graphics"
iIVBGM0GT1 = "Intel(R) HD Graphics"
iIVBGD0SRVGT1 = "Intel(R) HD Graphics"
; HSW Classic
iHSWGT2D = "Intel(R) HD Graphics 4600"
iHSWGT2M = "Intel(R) HD Graphics 4600"
Beta INF:
DiskId = "Intel(R) Graphics Media Accelerator Driver for Windows"
Intel = "Intel Corporation"
iIVBGD0 = "Intel(R) HD Graphics 4000"
iIVBGM0 = "Intel(R) HD Graphics 4000"
iIVBGD0GT1 = "Intel(R) HD Graphics"
iIVBGM0GT1 = "Intel(R) HD Graphics"
; HSW Classic
iHSWGT1D = "Intel(R) HD Graphics"
iHSWGT1M = "Intel(R) HD Graphics"
iHSWGT15D = "Intel(R) HD Graphics 4400"
iHSWGT2D = "Intel(R) HD Graphics 4600"
iHSWGT2M = "Intel(R) HD Graphics 4600"
That pretty much sums up that. Intel support tried to tell me that I had bad hardware and to replace it. I then told them that I bought 3 sets of this hardware and they all do the same thing. They said they would get back to me in 24-48 hours. That was last Friday.