You can book CAG-BCN-MIA on (VY to AA) on Iberia's site then a separate ticket to MIA-TYS on AA. Obviously not fully protected but maybe that is more acceptable if you want the VY option.

If flying OW/AA was important, personally I would fly early morning CAG-FCO on one ticket, then FCO-TYS on AA. Plan a ~3-4h layover at FCO and I think this is pretty low risk.