I agree, it sounds like there is more than a cooling problem. The radio should not be dropping power based on heat.
I also own an FT-840, I've never noticed the radio getting too hot. The fan seems to come on as necessary, including when I run SSB. Your problem definitely sounds related to a thermal issue.
I remember reading a similar complaint a long time ago that someone had with an FT-840, which was related to a component in the band pass filtering breaking down under heat, but I just cannot remember the exact content of the information.
Nevertheless, I seem to remember the person saying the circuit became mistuned when it got warm, and it would cause the protection circuit to kick in.
Does it have the same problem on all bands?