What GPU does the HP Elitebook have? The GPU needs to support DSC to achieve 5K60 from a non-Thunderbolt computer. Without DSC or Thunderbolt, you will need to create a lower refresh rate timing to achieve 5K. An LG UltraFine 5K can do 5K39 from DisplayPort. The Apple Studio Display might be able to do 5K47 from DisplayPort.
Did you test the USB-C port of the HP Elitebook with any other displays?
A USB-C connection would be the best option. Does the USB-C (Thunderbolt 3) cable work with a USB device (external storage) that can do more than 5 Gbps? A Thunderbolt 4 cable might be a better option since it’s guaranteed to work with non Thunderbolt devices. I suppose a passive Thunderbolt 3 cable should also work but I’m not 100% sure.
With the CAC-1336, it may help to put a Thunderbolt 4 hub between the CAC-1336 and the USB-C display, but the Studio Display is like a Thunderbolt 3 hub already so it might not be helpful in this case but if you have such a hub lying around, it won’t hurt to try it.