Dot 3 brake fluid is the lowest quality you can get, synthetic or otherwise and was really designed for older vehicles. It has a very low boiling point and in a high performance motorcycle your fluid could get very hot. Also it is very hydroscopic (absorbs water).
Suzuki recommend Dot 4 fluid and they also recommend never to mix fluids ( although Dot 3, Dot 4 and Dot 5.1 can in theory all be mixed). Dot 5 is not recommended and should never be mixed with other Dots) Dot 4 has a higher boiling point and is slightly less hydroscopic. Assuming your current fluid is Dot 4 non-synthetic, then if you are going to a full synthetic, you should completely flush the system before changing. But full synthetic isn't really that critical. It is recommended for ABS braking systems as it has a lower viscosity (is thinner) than non-synthetic.
I would buy a good quality Dot 4 standard fluid or if you wish, a synthetic Dot 4 but as you cannot be 100% sure what type is already in there, it would be a good idea to flush the system anyway. I'm sure the fluid you already have in there will have collected some water moisture anyway which can corrode the brake lines over time so a change as opposed to a top-up is probably best.