How to Record Screen in 4K on 1920x1080 monitor? (OBS Studio)
The easiest way I think would be to set up a virtual machine. I use VMWare Workstation 12. However Virtualbox is free. Youd have to install your OS and screen capture software (such as Bandicam which I use and recommend) on the Virtual Machine. 

But you could run the VM at 4K resolution and use it to record at 4K even on a 1920x1080 host machine without issue provided you have enough RAM and CPU power to do so. (Most machines should).  

Another upside is you could minimze the virtual machine while its capturing and still use your computer without interfering with your screen capture.