I think the problem is more the very early flight to LHR, for which KJ isn't open (it says 05 hrs, it's actually later than that). Frankly you'd be much better advised to spend more time in bed, even if it was open. In any event, there's no easy way to get to the Club lounge if flying BA, though they are both landside, the somewhat odd building structure at HAJ gets in the way. The Club lounge, which is a Priority Pass lounge, is on floor 3 above the check in area of terminal C. But definitely not open to BA customers, it's a third party lounge and BA won't pay the lounge fee.