I'm more of a UCCE / CVP guy, but I'm pretty sure this is not possible with UCCX as CUCM terminates the SIP conversation and UCCX just gets JTAPI messages, which wouldn't include SIP headers. Probably not really an option, but with CVP 8.5 you have complete access to all the SIP headers in your UCCE scripts so there it would be possible.
Now this is truly a very 'creative' solution, but if you desperately wanted it you could could parse the SIP headers on the ingress gateway with a TCL application, send them to a backend application using HTTP together with the ANI or call ID or so for identification and then pull them out again from a UCCX DB step. Would be a LOT of work, but it's possible in theory.
Cheers,
Kris