I don't know what the specific limits of the FCR are, but an AB class DDG has (3) AN/SPG-62 FCR's, and a Tico has one each AN/SPY-1, AN/SPQ-9a, and AN/SPG-62. The ships in the CSF are NetFORCE'd, so they are sharing data and presubably tracking and targeting tasks. So you have 2-3 AB's and 1-2 Tico's minimum, and if there is another CSF within the NetFORCE range, double that. On a wartime footing, they will not be operating alone. Add to that the other ships in the force, aircraft, etc, and consider that the attackers are themselves under attack, as well as their bases.
My thoughts are that the biggest threat to a CVN would come from an SSK operating either alone or in pairs, trying to sneak inside the screen using dash and drift tactics. Try to get in front of the CBG and then just lurk and wait for it to come in range. I don't know if they could actually sink one, but they might be able to disable it before being discovered (if they were reeaallly sneaky).