I think the concept of ASW carrier is awesome on paper, but it runs into several problems:
1) The carrier itself is a high valuable unit. It can't escort itself, because the usual armament of carrier is for point-defense only. For ASW duty, landing ships can be scratched from group roster, but a proper escort(s) must stay.
2) Helicopters used for ASW can't defend itself against an air attack. Air supremacy will be necessary for undisrupted ASW work. Against an opponent who has a functional air force, you need to provide an air cover for your helicopters. Who will provide the cover? There will have to be a friendly supercarrier, or an airfield in the vicinity. The half of a dozen of F-35B would be hard pressed to deal with a concentrated attack.
3) Also, some kind of AEW is necessary to warn against an imminent air or cruise missile attack. The first downed helo is not enough warning. The F-35 can run CAP, but the ASW carrier will have only a limited number of them. AFAIK the smallest ASW airplane in the US inventory is E-2, which can't take off from an ASW carrier. Fortunately, other countries have helicopters with air search radar (UK, Spain and China come to my mind).
4) Short range surface-to-air missiles poses no threat unless the ASW is done very close to the shore. Long-range SAM systems are something different. They have to be neutralized beforehand. The stationary sites can be taken down with cruise missile barrage from escorts, but the mobile ones will be much harder to find before they shoot.
5) An increased amount of VTOL aircrafts can mitigate some of the above mentioned points. However, the carrier deck is not inflatable. If you increase the amount of fighters for air cover (or other tasks), you have to decrease the number of ASW helicopters, which will limit your primary mission.
My experience with VTOL aircraft carriers usually goes into two ways - either the opponent can't defend itself and the carrier is a serious overkill for the situation, or the opponent can effectively fight back and then the situation goes quickly to hell, because I overestimated the offensive power of my carrier. I think the scenario balance can be challenging, but doable.