Can this circuit suffer from short-circuit?
You should really include a proper schematic of this circuit, as it is now it is difficult to see how it works. I have my doubts about this circuit as well as there is no base resistor present for the BC547 transistor.

Regarding short circuiting:

There is always a danger of a short circuit, especially with children as you cannot predict what they will do.

Instead of trying to prevent a short circuit, I would make the design such that a short-circuit doesn't cause any damage.

The problem with a short circuit is that too much current can flow. If you simply prevent that from happening, by limiting the amount of current the battery can deliver, a short circuit should not cause any problems.

It all depends on what type of battery you intend to use.

Since this is a low-power circuit you don't need high power batteries. I would choose Zink-Carbon batteries, so not the Alkaline type. These Zink-Carbon batteries also happen to be the cheapest.

Zink-Carbon have a reasonably high internal resistance and that limits how much current they can deliver. If you would use a 9V block or a couple of AAA batteries in series, hardly any damage can be done.