WooHoo another ADOT!
HI BadgerTom.
Change the gear oil. Gets rid of any moisture. Also if you have a moisture leak into lets say the drive shaft then you will find it when the oil you are changing comes out milky (contaminated with water) in which case there is something else you need to fix.
So changing oil in the tranny, drive shaft and final drive 1)removes moisture, and 2) helps identify trouble in a timely manner.
But if you'd like to leave water in your gear box, drive shaft and final drive then go ahead. But have you priced replacement parts for those bits recently..

Btw - I change out the gear oils
twice a year when I change from summer to winter oils in the fall and when I go back to summer oils in the spring. But then I ride a lot and my bike lives outdoors. But then again I'm at 130,000 miles on the original gearbox so I must be doing something right. 8-)
Now on the matter of your synthetic engine oil - Yes it is a better oil than the Dino oil I use and Yes you can let it ride for more than a year with low mileage. Because it wont let the acids build up that will start to cause corrosion. But keep an eye on your oil consumption and look for leaks. 1980 was the tuning point for Oil Seals. After 1980 all seals made had to be compatible with synthetic oils. Before 1980 ugh Good Luck. I have had experience with this when I tried Synthetic oil in my 03/1980 bike's gearbox and was rewarded with a nice clean twelve inch puddle of really clean expensive gear oil on my garage floor underneath the shift shaft seal.
And in true keeping of every ADOT I will now tell you what oils I use:
Engine - Summer Valvolene VR1 20W50 - Winter Valvolene 10W40 Motorcycle oil. Both of these are Dino Oils with lots of anti-wear Zinc (ZDDP) that I can buy at Autozone when I am traveling about the country.
Gearbox, Drive shaft, Final Drive - Summer Valvolene Durablend SemiSynthetic (Dark Blue Bottle) 85W145
- Winter Valvolene Durablend 80W90.
I'm in Atlanta GA and I ride year round. Your results may vary.
Regards,