If anything - Shouldn't a dedicated (i.e. designed-from-the-ground-up) ULRSAM be much more capable at engaging aerial targets than a cruise missile with surface strike as its primary mission, instead of the other way around?
Honestly, the use of CJ-1000 in combination with the AEW fleet to prosecute the B-52 and C-17 makes sense. Especially the B-52 currently provides most of USAF GSC's strike capability against China. In all domains, bombers are special because they have speed, range and a decent payload. The B-52 is particularly special because the B-1 fleet is a mess because of their past use for CAS in Afghanistan and the B-2 fleet simply lacks the numbers and payload capacity. The C-17 is also getting the Rapid Dragon for a bomber-like use.
If I come to the CJ-1000 itself, it physically should have the capability. Software and command system implementation would be development challenges.
Speed: At Mach 7 plus it can definitely catch-up to the bomber. The B-52 would launch its JASSMs from about 800 km off the PRC shores. At an average relative speed around Mach 6, it takes about 9 minutes for it to cross the distance. Mach 7 is also faster than most AAMs and SAMs.
Altitude: All hypersonics fly high at 20+ km altitude unlike subsonic LACMs.
Seeker: It would have a large AESA. Modern radar behavior is software defined. The tasks that require dedicated aircraft in the past are just software packages in 2025. An air search function could be implemented. 9 minutes mean a lot of displacement. But the B-52 and C-17 have massive signatures and the missile will have a datalink anyway.
Maneuverability: This is a wildcard. China loves maneuverability in non-counter air missiles. The YJ-18, YJ-31, HJ-10, etc had their maneuveability marketed. I bet the CJ-1000 was designed with significant maneuverability to increase its chances against the Patriot and SM-6 et al. Which might be enough for bomber interception.
To sum up, I believe the proposition is viable.