I would try AS outbound coming from SFO. Ive never been denied even on lite fare when connecting onward to CAI (im here right now). However on lite fares on return, it depends on the agent. There's one I know who will absolutely refuse it. The others normally if you hand over the CAI-DOH BP will simply scan, stamp, and on you go. Technically no access should be given as it is a through P fare that simply books into F (There's no J on intra-ME flights) on DOH-CAI and v.v.