My biggest difficulty with an alt-Falkands war is the use of the acronym FAA.
The Argentine air capability was not going to sink all or even most of the task force. However, it is certainly within the realm of possibility that it could have damaged (or maybe even sunk) one or both carriers, which would have ended that attempt and caused the task force to withdraw out of range of further air strikes.
On the other hand, the Argentine navy was not really capable of conducting effective ASW operations against the RN SSNs as everybody else has already noted. Supply of the islands (whose garrison was ten times as large as the local populace, when considering the demands on the infrastructure there) would be quickly restricted to C-130s only. Type 42 destroyers are AAW platforms; they are barely capable of self-defence against a modern submarine in blue water, lacking a towed array. The other Argentine vessels were equally not fit for ASW against a front-line power - there were a few French A69 corvettes which were the least-bad but still with very short-ranged sonar and weapons compared to a Churchill-class. The remainder of the escorts were WWII USN destroyers, with WWII-era sensors. In terms of maritime patrol aircraft, the S-2s were still fairly capable, and could provide some air cover for supply convoys, but I don't know what ANA stocks of sonobuoys were like - I would doubt that they would be able to maintain that sort of patrol schedule for long. At that point it's down to accidentally flying directly over the sub and getting a MAD contact. There just isn't that much ANA sealift capability - losing three or four transports (one of which was sunk by a Type 21 with gunfire, IIRC, though that would probably be butterflied away with an early Exocet success against the CVHs) would again jeopardize the ability to either supply or withdraw the garrison.
If the Black Buck raids might be tried with a bit more success, or heliborne commando raids attempted on Stanley airport or the desalinization plant (though that was rejected earlier by the RN as a bombing target), the supply situation for the occupation force might well become completely untenable.
As to ally operations - IIRC, the biggest threat to Argentina was Chile; Galtieri had not managed to keep secret that his next trick for keeping the junta in power was going to be a short, victorious war with Santiago, but I've got to admit that anything I would propose there would just be idle speculation on my part.
Thinking about it, Argentina would have had an absolutely terrible time with the RN's SSKs, which were really very quiet for the time - did the RN have the capability to forward-deploy the Oberon-class, with RFA support? South Georgia would be best; I think the travel time from and to Ascension would leave a very short patrol endurance. Worse would be the possibility of mining the Argentine ports, although that would be an escalation that the Thatcher government would be hesitant to embrace.
While the Marine Nationale with Foch and Clemenceau would probably have been a decisive factor, Thatcher would not have wanted to ask, and Mitterand would not have wanted to be asked, for that sort of help. Probably electorally fatal for the Tories, and I really don't see Labour at that time joining any national unity government for the duration of hostilities - even though Foot might well have put partisanship second, I don't see Benn letting that chance slip away.
I don't think the RN could man a Kitty Hawk in a sensible time frame, to be honest. Either a lot of USN personnel would put on new uniforms, bad Dick van Dyke mockney accents, and revel in a service that serves beer, or a solution will be reached via submarine blockade before that.
The worst case scenario for everyone, is that Galtieri and Anaya decide to have all of the Falkanders thrown out of the back of C-130s out over the South Atlantic. Then Thatcher can escalate with USN and MN support freely.