So, can we assume your friend who builds tube amps has done the following?
- Pulled the cover and blown the dust out
- Cleaned the contacts on the effects return jack
- Checked for loose/corroded connectors (spade, molex, etc.)
- Checked for cold/separated solder joints
- Checked for loose PCB/standoff mounting screws
- Fan filter (if there is one) clean
- Fan working
"On for a few minutes" isn't going to build up enough heat that an oversized heat sink would make a difference. And besides, the amp ran all those years with the heat sink it's got, right?
If a problem occurs only after an amp is warmed up, then oftentimes a tech will have to bench the amp under a heat lamp, because the cover is off. However, on an amp where airflow is tightly controlled, an amp can actually run hotter with its cover off because the tunnel's gone missing (unless the tunnel is a separate piece that can still be left in place during testing
and the fan still has exclusive access to it, i.e. w/o air leaks and loss of pressure in the tunnel).
If it's none of the above, then you might have a component that's giving up the ghost. Pure guess here, but I suppose it's possible that the heat-removal scheme on this amp is insufficient, or someone ran the amp with a dead fan, with one or the other allowing the output transistors to fry or their thermal compound to harden and lose thermal conductivity. One serious overheat can change the characteristics of a power transistor, often resulting in its always going into thermal runaway, even at amp idle, but this is 100% conjecture on my part.
Anyway, I'd run that list above past your friend and see if he did all that. It's usually best to proceed in a logical & orderly fashion. It would be unfortunate to upgrade the heat sinks only to be faced with the same symptoms.
Nice amp, though. Good luck, hope you get it resolved.
Sunn 1200S Hyrbid on the fritz... 
