What is the practical limit/best practice for the average rectified output current of a diode in AC applications?
Disclaimer: you're modifying a UL (or CE) labeled device; if you burn down your house or electrocute your dogs it'll be on you.  If your insurance company is particularly asinine, then if a Mad Bomber tosses a Molitov cocktail into your bathroom window and burns down your house, they may use the presence of your un-plugged but modified hair dryer as an excuse not to pay.  Because -- insurance companies.

From the sheet that you link to:


  In the absolute maximum ratings, one or
  both of the following are given:
  
  
  The maximum permissible average rectified output
  current for zero diode voltage (reverse).
  The maximum permissible average rectified output
  current for the maximum value of VRRM during the time
  interval at which no current is flowing.
  


So, find a diode that specifies the second, and use it.

In practical terms, the diode heats up mostly proportionally to the average rectified current, because the drop is fairly constant, and power = current * voltage.  I believe that the reason the number depends on the reverse voltage is because the part's reverse breakdown voltage will drop with increasing temperature, and higher current = higher dissipation = higher temperature.

Note, too, that the given ratings assume a certain ambient temperature, and a "typical" installation with free air circulation.  If you use your 3A rated diode, but put it into the same narrow, no-airflow space, its junction temperature may get just as high or higher than the "1A" diode.