First post, thanks for this wonderful forum. 
Sorry for digging this up but I had the exact same heat problem and I solved this issue with a small variable power regulator module that takes almost any input from 4.5-28V and can regulate it to anything from 0.8 to 20V (as long as it's less than the input; it's no step-up converter / inverter). What you do is to connect the input of the power regulator module to the unregulated + and the GND as shown in irony7's post with one wire each. Since the DMG's switch switches this power source, your backlight will actually turn off together with the DMG. 
After having soldered the module to the power source and before connecting the backlight you switch on the DMG, measure the output voltage with a multimeter and regulate it to 5V with the module's potentionmeter using a small flathead screwdriver. After that you simply connect the backlight's wires to the output and put everything back into the case. After this you basically have two parallel power regulators, the original DMG's and the new one. It works like a charm even with the EMS flash cartridge. You can easily find these regulators by searching for "DC Buck Converter Step-Down Module" at your favourite online store. I got six of these for 9 €. Make sure you buy small form factors.