Hello c.riley,
Please note Unreal Engine 4 currently does not import 16-bit images in sRGB colour space.
To have matching colours, you can export your Base Colour in 8-bit depth. To achieve this, you can set the 'Output Format' of the last node before the output to 'Absolute', and its setting to '8-bit per Channel'. The text below the output should read '2048x2048 - C8'.
You will still need to check the 'sRGB' checkbox in the texture parameters in Unreal Engine 4.
For your information, the Usage settings in 'Properties > Integration Attributes' has no effect on the exported bitmaps. The role of these settings is to make sure the outputs are connected to the appropriate samplers of the shader used in the 3D View.
Feel free to get back to us with the results of following these steps, or if you need further assistance!
Best regards,
Luca Giarrizzo
Note: Since your issue is not connected to an export problem, we went ahead and changed its title to more accurately represent the issue.