While the processor itself supports very low-power modes the LM2734 Datasheet shows a typical quiescent current of 1.5mA. It is also only a step-down regulator so the Arduino Due shows a minimum operating voltage of 6V, but you'd really want to aim higher say by using 6 cells for 9V so it continues operating once the batteries are getting low.
If you used 6 x 1.5 AAA after a quick search the highest capacity I could spot was 1200mA/hr so that would give around 1200 / 1.5 = 800 hours or about a month of operation.
But that's only taking the main regulator into account, there's also op-amps and a secondary linear regulator along with a few resistor dividers. I do not know the exact answer but as a guestimate would say it's likely to be more in the order of 5mA (10 days of operation) and not really suitable for your intended purpose.
You shouldn't have to rewrite the Arduino Due bootloader - you just need to write a program that can act as USB HID using the native USB port, and upload the program to the Arduino Due. You can use the Due's programming USB port to do the upload.
But here's info on the Arduino Due boot process:
The Due has a ROM bootloader that listens to the USB native port and Serial port 0. You can talk to the ROM bootloader via USB directly by resetting the SAM3X8E and using the native port. But I've not found that method to be very reliable. The more frequently used programming port is connected to an Atmel AVR 16U2 microprocessor acting as a USB-serial converter, connected to the SAM3X8E serial port 0. The Arduino 1.5 IDE can talk to the 16U2 using the Bossa command (bossac), reset the SAM3X8E, and upload new programs to it it. This page explains the Arduino Due's bootloader in detail.
If you're interested, here's the source code for the 16U2 AVR USB-serial converter.
And here's Bossa - the Due's command that corresponds to avrdude.
Using bossac (the Bossa commandline tool), you can upload code the SAM3X8E, using the builtin ROM bootloader.
On creating HID devices:
Here's a link to an Atmel application note that has example code for creating a program that can act as a USB HID keyboard:
http://asf.atmel.com/docs/latest/common.services.usb.class.hid.device.keyboard.example.sam3x_ek/html/index.html
The source code to this example is the in the Atmel ASF packages - unfortunately, a large download. Here's where to get it:
http://asf.atmel.com/docs/latest/download.html
Once you download it, the source code the HID keyboard example is in this folder:
common/services/usb/class/hid/device/kbd/example/sam3x8e_arduino_due_x
Here's a list of the other app notes, with examples of other kinds of USB and HID devices. Look for the ones that are for the Arduino Due:
http://asf.atmel.com/docs/latest/applications.html
Best Answer
The datasheet for the Atmel ATSAM3X8E Microcontroller used in the Arduino Due specifies the following about bypassing and stabilizing VDDOUT:
From Table 46-3 of the "SAM3X/SAM3A Series Complete" datasheet
The capacitor value specified for stabilizing / decoupling a power input or output pin in datasheets is typically a minimum, indicative value - If there is also a maximum acceptable value, that may conceivably be separately specified.
Electronics designers often use a higher value capacitor within the same order of magnitude, if they foresee a high load on the power rail - The higher the capacitance, the better it is able to weather out load-related ripples on the power rail.
The upper practical limit on this capacitor value is set by the initial power-on current that would be drawn from the power pin to charge this stabilizing capacitor - Too high, and the supply may be damaged.
The (optional) 100 nF capacitor specified for bypassing, is for the purpose of providing a short-circuit path to ground for any high frequency noise on the power rail. This is placed in parallel with the stabilizing capacitor, both as close as possible to the power pin concerned.
That is why you see the 10 µF and 100 nF capacitors in the Arduino Due reference schematic.