The lid should say dot 4 only. And you should use dot 4 or higher. If you put dot 3 in there you might have to drain it all out and flush/ bleed the brakes. It wont handle the high temps of a sport bike. You don't want your brakes to fail.
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.
Motul RBF600: Excellent Dot 4 fully synthetic fluid. Used by many racers and has an extremely high boiling point and very low hydroscopic properties. Some may say it's overkill on a road bike but I agree with you da man.....Anything else is a compromise!!
I would say yes...Your brakes are probably the most important item on a high performance motorcycle, it's prudent to make them as good as possible. Dot 3 is,to be honest, crap and not really any good for a sportsbike. Suzuki recommend Dot 4 and not to mix different Dot fluid. Although this isn't essential, Suzuki recommend it to ensure maximum performance and remove any chance of compatibility issues between different Dot fluids.
It's up to you at the end of the day and carrying on as you are, you may never have a problem with the brakes, but wouldn't you feel safer knowing your brakes are as good as they can be?
Motul RBF 600 is one of the better racing brake fluids for track and hard street application. Though it should be changed with some frequency, depending on usage. Fluids with high dry/wet boiling points tend to be more hydroscopic. When in doubt, flush the old and refresh.
I would stick to the same brake fluid rating. Be it DOT4 or full synthetic. Don't mix em together at all. I'd recommend doing a full brake bleed at the end of each season or at some point during a given year. I mean it's fairly cheap to do!