Somewhat, this depends on the source: is that a regulated, or just a filtered 9-12V?
To shed heat outside the regulator, you could either put a resistor between the
source and regulator input, or a resistor/capacitor parallel pair, if surges are expected.
Some three terminal regulators are quite sensitive to any input
resistors, since they start to oscillate. To eliminate this, a big
capacitor directly at regulator input to ground is required.
Put the resistor/diode/choke between the rectifier bridge and the main
reservoir capacity, so the capacitor is next to the regulator input.
A 1W diode and its mounting/heatsink is more expensive than a 1W resistor and its mounting.
The voltage drop on the resistor has a higher dV/dI than an equivalent string of
diodes, though.
Thinking outside the box, you could go to a lower-current regulator and a PNP transistor,
with some current-sharing resistors, and move most of the heat dissipation into the
transistor (assuming the transistor can take higher temperatures than the IC regulator).
There's also the possibility of adding a choke to the AC into your source rectifiers, which
will drop V without dissipating power.
Choke input power supplies (or C-L-C) were popular in the tube
rectifier era, since tube rectifier could not handle huge peak
currents into a simple big capacitor. The choke in L-C or C-L-C
configuration extended the conduction angle and the average DC voltage
was close to the secondary RMS voltage.
Unfortunately the choke at 100/120 Hz needed to be big and heavy and
needed to also handle DC current (air gap), so it is understandable
that chokes are seldom used these days.
