This might seem silly, but I've had issues with this before with client's computers and I resolved it by simply refreshing the Windows Experience Index.
With that being said, your resource monitor seems to be detecting everything correctly and shows that more than 2048MB are actually being utilized, thus the new stick seems to working despite the fact that task manager doesn't recognize it.
I believe Vista/Windows 32-bit versions also will report the full 4096MB as shown by your screenshot, but will only truly utilize 3.5~ like you had said.
Okay all of the RAM seems to be working, and the majority of the RAM as I thought is being used in BIOS for the iGPU.
Simple, boot into the BIOS and change the iGPU Memory to something less then Save & Exit = Yes. Also, with a 32-bit OS you'll only be able to access between 3.0GB to ~3.5GB of RAM; this is an OS limitation. There are Apps that can recapture that loss but I really don't recommend them.
To be honest I'm not sure, if you don't have a discrete/dedicated video card with its own memory, that sharing will likely be dictated at the hardware level. If that is the case, it will likely be located within your BIOS.
Someone else may know something I don't though so don't take that as the end all.
The vast majority of the 1282 MB is dictated in the BIOS as 'iGPU Shared Memory' since you didn't list your MOBO I have no way of telling you exactly -- since every MOBO's BIOS is different and can call it something else.
Again. boot into your BIOS and change the value. IF again IF you have a Discrete GPU and your monitor is connected to it then use the lowest value e.g. 32MB~64MB and/or Disable if it's and option, and then in the Device Manager Disable the iGPU and reboot. This will reduce the shared iGPU memory to it's lowest possible value.