A hypothetical air launched ULRAAM that is equipped with a scramjet engine would be far more useful than applying the CJ-1000 in a SAM role.
The CJ-1000's primary role would be anti-surface. SAMs and surface to surface missiles have very little overlap in booster, warhead and guidance (and any other internals) optimisation. If you dont have an issue with the speed, control and guidance of a HCM to hit a large and slow but still moving target in the air, then you can make a missile that is more optimised in the anti-air role. You wouldn't make it out of a anti-surface missile.
Unless there is another variant of the CJ-1000 family, a HQ-1000 applying the same technologies but optimised for anti-air, the CJ-1000 should be considered a surface to surface missile only. If you had all that tech, just make a H-6 or J-16 and in future, J-36 and H-20 launchable ULRAAM that applies scramjet. If there is a hypersonic ultra long range anti-air missile whether it is in SAM or AAM form (or both), it would not be the CJ-1000. It'll simply apply similar tech. So why we continue thinking of the CJ-1000 as this/these hypothetical missiles is strange. A DF-26 with MaRV can potentially be used to hit a AEWC aircraft too. We shouldn't consider a DF-26 a SAM if a MaRV SAM is a workable concept.
I've no doubt at all that PLA is interested in fielding a hypersonic, air breathing AAM and SAM. We should just call it as such since the fundamental tech required for this is all there. If they add a booster large enough, these missiles could have 2000km+ range.