My 60D has never experienced overheating, even shooting in direct sunlight in 35 degree celcius weather.
The 60D is only a few dollars more than the T4i at the moment so is most likely still within your budget. The T3i also has much less overheating issues than the T2i, though I cannot personally vouch for it, only going off what I've been told or read.
I would suggest you get either the 60D or T3i over the T4i; it does not add any really useful features for video - the continuous AF is pretty useless, and even the touch focus feature is not acceptable for use while recording.