I've read in some texts that GW-BSE quasiparticle calculations have only been implemented on Norm-conserving Pseudopotentials (and not Ultra-soft or PAW PPs). Is there a conceptual reason for this? I can't seem to locate a reliable source but here it is mentioned that the YAMBO code (used for GW-BSE in conjunction with Quantum Espresso) can only support NCPPs.
The GPAW code can do GW/BSE and it uses purely PAW potentials. So it's absolutely possible to do GW/BSE with PAW sets. The involvement for the implementation, however, is another topic. From Kevin J. M.'s answer, the primary block is fairly clear: PAWs are quite involved to implement. Doing all of the atom-centered corrections and then getting them onto your regular grid requires a focused effort to do, and is perhaps not the highest priority of many codes out there (somewhat understandably so, but PAW does allow very low PW cutoffs...).
In case you're interested, the tutorials for GW/BSE in the GPAW code are as follows:
I'm fairly confident that Abinit also supports PAW potentials + GW/BSE as well. I think the key point is always how much the PAW methodology is prioritized by the code's developers.