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Unfortunately, your problem lies primarily more in how the speed bumps you're encountering are constructed, rather than your vehicle itself. In an ideal world, these speed bumps would not exist at all, as we would all know how to drive appropriately under all driving conditions, but that's not likely to happen here.
According to most accepted standards, a properly constructed speed bump should be at least 3.66m in length from end to end including the slope, and shouldn't exceed more than 10.16cm in height--with 6.35cm being the nominal height. Speed bumps are supposed to be primarily used to slow down traffic on lightly traveled streets and roadways, not busy streets. Unfortunately, that standard is hardly followed here.
Here, speed bumps that are more aggressive in nature are more common. Also, speed bumps are supposed to be used primarily for entrances to private roadways and parking areas, and not public roads with high traffic. If you look at the document measurements of speed bumps, you'll see that they aren't car-friendly at all. Their specifications range from 0.3m to 0.9m in length, and anywhere from 7.62cm to 15.24cm in height. Because of their short lengths, these humps are the least vehicle-friendly of all the speed-limiting obstructions, and you will damage your vehicle when you go over these at speed.
The limit of most cars' ground clearance is 15.24cm. To address your questions, there are a few basic ways to increase ground clearance. One way is to change your tires to a set with a taller outside diameter, just as you mentioned. Doing so will increase your ground clearance by half the amount of the increase in the new tire?s diameter. It is possible to do this on most cars by a small amount (as little as half an inch to as much as two inches on some cars), but it comes with the disadvantage of effectively changing to a taller final drive ratio. The net effect is your car's acceleration will be slower, and the overall feeling is that your car has lost power or become slower in general. You would also need to do this to both ends at the same time, otherwise one end would be lower than the other as you also already mentioned.
Another method is to change to a suspension with one that allows for a taller ride height. This can be done through the use of an adjustable height coilover suspension, or an aftermarket air suspension. Both of them will be costly, and they also come with their own disadvantages.
To achieve the increased ride height for adjustable coilover suspensions, the suspension geometry will deviate somewhat from its original design, and it's possible that you will go overboard in some cases. An inch to an inch-and-a-half higher is still within acceptable limits for most cars. If you go overboard with the height increase, however, there will be adverse effects on the handling. The parts needed to do this will also likely have to be custom-ordered parts.
Air suspensions, on the other hand, allow you to raise or lower the vehicle by a certain amount to clear the obstruction. But that would assume that you will notice the obstruction, stop and change the ride height before going over it. Or just keep the vehicle at a certain height all the time. It's a a lot more expensive than coilovers.
Polymer spacers inserted into the coil springs are probably the cheapest solution for you. They reduce the suspension travel by a certain amount by preventing the springs from compressing fully. While this does not change the suspension geometry, it does affect ride and driving characteristics somewhat, particularly under conditions wherein there is significant suspension compression. They are certainly cheap enough for you to try them, and not feel so bad if they don't work.
All of the aforementioned methods will likely void your warranty, as they affect the normal functioning of the suspension--particularly the ones that require any sort of parts replacement. The most affordable would be the polymer spacers, costing as little as 2000/- for all four corners. They're also the least intrusive among all the options available. Personally, I would explore that route, then the coilover route, before considering the more expensive air suspension modification.
TG
