Also note that at that link you posted, they note that the Uno with the software serial connection doesn't do well at the 115200 baud rate the servo, by default, expects. This is because a software serial connection can't maintain this speed (for a few reasons). Instead, they recommend that you use the lower speed of 57600 baud (you may even need to go lower - the software serial library really is only reliable up to 9600 baud; to go higher you have to be careful with your cabling, lengths, etc).
The Mega (which they also have examples for) can go higher because the pins they are using tie into one of the other three hardware serial ports on the Mega, which can support the higher speed of 115200 baud.
Your problem (I think) will be that in order to switch the baud to the lower rates, you need to have some way of communicating with the servo at its current default baud rate (115200 baud).
So - you will either need a Mega - or you'll have to get creative.
I noted that they don't provide any examples for the Due - likely because the Due is fairly new, plus it doesn't have the popularity of the other boards, and it is a 3.3 volt device (meaning that you may need level shifters to get it to communicate with the servo without damage to the Due). That, and the Due may use different pins - I'm not sure if it has other hardware serial ports or what (if it does - and you can hack the driver for the servo to support the Due, and you can verify whether you need level shifting or not - that might be one way to re-program the baud rate of the servo).
Another way - if you can reliably "code blind" (or do a lot of trial and error testing) - would be to using the Arduino's hardware serial port to communicate and change the servo's baud rate to something lower. You might be able to use LEDs on the other pins for status/debug purposes; or if you have some kind of TTL serial to USB serial device - use a software serial connection @ 9600 baud back to your computer (via USB) to monitor it.
Basically - you would want your sketch to set up the hardware serial port to the 115200 baud rate, then wait for a button to be pressed. When it is pressed, send the commands to the servo to change its baud rate. At each step along the way, blink the LED, or print out (to the software serial port) some kind of status message so you know things were successful. Once you have the Arduino loaded with your sketch, disconnect it from the computer, then hook the hardware serial pins (0 and 1) to the servo, then press the button and monitor the status. If/when you get the "all clear" - then reflash the Arduino with the test code you have, but set the baud to the lower rate (whatever it is), and try the example.