I have over 700 hours and much IMC on my 4-P and ice with associated hot engine problems is the way it worked for me also.  I have had over an inch of ice on my wings and had to power back to under 20 inches MP to avoid overheating and it really gets your attention, once my IAS went below 120 kts in the flight levels.  I have talked to one other 4-P owner with no hot engine problem in ice and we could never figure out why me and not him.  I did get a K&N filter and enlarged the opening into the air filter and that did seem to help by keeping my alt air door from opening.  It may be your alt air door needs work.  I did not have a manual alt air door control on the cabin and would have had I to do it over again.  The issue I don’t think is inlet cowl air but rather the induction air from the cowl into your engine when the alt air door opens after your air cleaner ices over.  I sold my piston and am building a turbine so I quit working the problem, but Dale Sholtz of Helena Montana has a 4-P with over 1000 hours and he has been iced up many times with no hot engine problems.  Another 4-P owner, Gene Hughes of Butte MT has the same problem as you had and I had, so go figure. 

 

If you had a manual alt air controller you could figure out if the problem is hot cowl air by manually opening the alt air door to see how it performs.  Once climbing out of LA I had hot engine problems and only very light ice on my wings, perhaps 1/8 inch.  That would seem to indicate its not the induction cooling air that is the problem, rather the ice covers up the air cleaner, the alt air door opens, hot cowl air then goes into your engine, and then you have the hot engine problems.

 

I experienced what you did many times, and its no fun.