If it's a good quality power supply, the pulsing means the power supply is possibly overloaded. It's the power supply's, method of telling you that, which is called "hiccuping." A load of 156W may be too close to 200W. You should allow for load + 20% when choosing a power supply.
First I would go with warm white lights not the cool white, I think it makes a softer look to the bedroom.
In terms of the switching, yes a push button switch would work fine, you just have to make sure that the switch is within spec for the lights used. I have rather opted for a dimmable solution with my childrens beds. since they like to have some light at night. a good option here is this:
![enter image description here](https://i.stack.imgur.com/iWza1.jpg)
Its rated up to 8 amps, so if you look at the strip in your post, it consumes 2.5amps per 5m section used. (provided that you buy a transformer that can supply up to the required amps (ie 2.5amps per meter) -- you can buy a smaller amp transformer, but the lights will not be optimally bright at the highest level.
Your switches are only rated 3 amps, which means you can run about 5 meters of your spec LED Strip. -- Also to note, I would go with a stronger LED strip, generally 1 amp per meter is a nice powerful light (then you would only need a meter per child)
Another thing to consider, is the use of the 3color led's whereby you can have a color chooser.. but that may be getting too complicated.
Your series vs parallel question: You must definitely hook up the two lights in parallel, but just accomodate for the increased amps which will be required. The lights will draw the amps needed so you cannot really blow anything (its the volts which will blow the lights)
Finally: I find that the higher power led strips get a little warm, (meaning that the sticky tape sides start to get less effective over time). would be best to stick the strip to a metal surface if you could choose (it would act as a heat sync)
I actually ended up using a Chinese paper lantern to put the led strip in, it looked quite nice.
![enter image description here](https://i.stack.imgur.com/bizTi.jpg)
Best Answer
Depending on the type of LED element and the driver circuit, you could either be fine or significantly shortening the lifespan of your lights. If the instructions or markings don't indicate that AC is acceptable then it's probably not.
However, a failure is not likely to be dangerous or cause a fire.
So you have to decide if the hassle of replacing burned out LED lighting later outweighs the hassle of giving it the proper power now.