When you do a Tools > Burn Bootloader with the Arduino IDE, it actually does two things:
- Set fuses according to the boards.txt definition for the board currently selected from the Tools > Board menu
- Flash the bootloader to the microcontroller
So it is possible to set fuses via the Arduino IDE's Tools > Burn Bootloader. However, if the fuse configuration you want is not present in an existing board definition, you would need to either create your own custom board definition, or else modify an existing board definition (likely the "Arduino/Genuino Mega or Mega 2560" definition in this case"). If you only need to do this one time, it might actually be easier to just set the fuses from the command line using the avrdude tool the Arduino IDE uses for this purpose.
You will need an ISP programmer. I don't have any idea whether the one at the link you provided works with avrdude. My favorite AVR ISP programmer is the 6 pin USBasp, which you can buy on eBay for $2.15 w/ free shipping from China:
As you may have learned from the issue thread I linked to in my previous reply, the USBasp does have issues uploading to the ATmega2560 when using the stock USBasp firmware. That limitation is solved by updating the firmware to the 3rd party version produced by members of the Arduino community. Even with the old firmware, you can use the USBasp to set fuses or burn the bootloader on the ATmega2560, no problem. You can also use an extra Arduino board as an Arduino as ISP programmer:
To make life easier, you can grab the avrdude command the Arduino IDE generates to set the fuses, modify it, and then run it from the command line:
File > Preferences
Check the box next to "Show verbose output during: upload".
Click the "OK" button.
Tools > Board > Arduino/Genuino Mega or Mega 2560
Tools > Programmer > select the appropriate programmer
Tools > Burn Bootloader
After the Burn Bootloader process finishes, scroll up the black console window at the bottom of the Arduino IDE window. There, you will find two avrdude commands. The one higher up in the output is the command to set the fuses. Copy and paste that command, modify it according to your preferences, then run it from the command line.