I dont need a solenoid from that picture. I'm searching for the smaller one and it looks like I've managed to find one using the link you've provided but it's 26 mm in diameter ) I will search for one to see if it will fit into the gearbox.
And I think Kuba T1000 uses a solenoid powered by the battery - he has the battery itself and the electronic control unit to adjust timings.
I'm new to elctronics. Can you tell me what changes will happen if I will use a 12VDC solenoid with 7V Li-Po battery? Will it even work?