Rapido uses an electronic (Hall Effect) "switch". To duplicate you would have to find the component they use. They work from 4.5V but I think that the Hall Effect switch will work from up to 15V. The way they have that switch wired, it has constant battery voltage supplying power. If your models do not receive power when you are not running them on your layout I'm not sure of the Hall Effect switch will power up in off, on, or in a random state (if that matters).
I dug up the Easy-Peasy unit I bought for experimenting. (I'm not a fan of battery-powered lights).
It uses 3V to power the LEDs. It has an ordinary (non-latching) reed switch. It also has a custom-made integrated circuit chip (IC) embedded on the circuit board (the IC die is bonded directly to the circuit board and encapsulated by a drop of black epoxy). The reed switch is hooked up directly to the IC and it simply is used to toggle some flip-flop inside the IC. One momentary closure of the reed switch (by waving the magnet by it), turns the LEDs on, another momentary closure turns them off and so on.
There are no external resistors on the board so the LED current is either internally limited by the IC, or since the battery voltage is 3V, there is not current limiting (since the Vf of white LEDs is around 3V).
Since that IC cannot be identified, it would be difficult to find for purchase (it might be specifically made for that circuit).