I'm surprised that nobody has suggested establishing the rating of the pitot heater (power in watts at a specified voltage), and from that determining the DC current it should draw when in good condition. Then, having done so, measure the actual current draw. Until this simple test is done you won't know if the heater winding has 1) partially short circuited - this can happen, and will lead to increased current draw but not necessarily failure to heat up), or 2) whether the circuit breaker is tripping at less than its design rating. Breakers can (and do) age, and their current handling will reduce accordingly. The manufacturers will usually list the number of maximum current "trips" before permanent damage will occur. There will also be charts showing the current versus time curves which will trigger the contacts to release. It's just possible that with "all the other crap" switched on, the DC bus voltage is dropping enough to reduce the current draw through the heater circuit, and prevent the breaker from tripping. But until you've established the basics the rest is simply guessing.
Obviously what (if any) of the above the O/P can do depends on the status of the aircraft in question - Permit or CofA...