Probably, in this heating question, there is not much difference between the IIa and the IIc. I do a lot of 50x60cm prints with both IIc and Ic, and use the 250 bulb 25 percent of the time. All I made sure about is to have correct wiring.
As I prefer the original opal bulbs over anything else, this question of (over)heating has been important to me. Perhaps more than anything else. I have a Durst L1000 wall mounted and use the very large opal bulbs, 200, 250 and 300 watt. With this Durst the light uses a mirror to reach the negative and this means longer exposures. Durst made a hole in the lamp-head into which one could connect a flexible tube to blow air in there. I have used this hole to install a small fan and covered this on the outside with a black box that can take air in. I use that fan while setting up and after exposing. Not during the exposure, although I probably could. This works fine. I frequently have basic exposures around 60 seconds, and then again that once or twice for burning. With the use of this fan, there is no problem.
For 135mm negatives the surprising solution came from using the Valoy II for the 50x60cm prints (I modified this enlarger for that purpose). Exposures are a lot shorter with the Valoy II compared to the Ic. This is mainly due to the bulb being much closer to the negative.
Perhaps the above gives you ideas.