Can I power a lamp from both the switched and unswitched halves of a split outlet?
You will probably be better off overall to keep things simpler than the scheme that you think that you want. Mixing up the wiring could be problematic as indicated in the comments of the two parts of the duplex outlets are on separate circuits.

A better way to get the type of operation that you desire is to turn to the modern smart technologies. One example would be to look at the Lutron Caseta line of smart lighting products. With this you could setup something as follows:


Replace the switch for the other lights with a Caseta smart switch.
Add a Caseta plugin light controller to the always powered outlet to give smart control for the plug in desk lamp.
Acquire two Caseta PICO remotes (small battery powered remotes).
Either use wall mount kits to install the PICOs at a convenient wall location OR get desktop pedestals for the two PICOs.
Pair the PICOs with the smart light switch and the plug in light controller to get the type of control you desire. As described in you question, you would likely pair one PICO just with the plug in light controller to give independent control of the desk lamp. Then pair the second PICO with both the smart light switch and the plug in lamp controller to achieve the ability to control both the lights and lamp from a single control point.


(Note that I have no affiliation with Lutron or Caseta products other than having fully smartized my home with these products).