Firstly, the net external force acting on the system (all 3 bodies) is zero, which means that you are free to use the laws of conservation of energy and momentum. You need to understand that the impulse provided by the string only affects the velocities of individual bodies such that the velocity of both the bodies along the string joining them are equal, which seems pretty obvious as the string will break if the velocities of both the balls are unequal. And since we have assumed that the string is inextensible and there is no friction between any surfaces, it is clear that the total energy loss in the system is zero, and hence you can apply the law of conservation of energy. You also need to understand that the loss of energy in an inelastic collision is due to the deformation of bodies and energy thereby getting wasted(in deformation and also heat loss, if any) and also getting stored in the bodies in the form of potential energy. But in your question, all those possibilities are sidelined due to the assumptions we have made (that I have mentioned above). I hope I answered your question, have a great day!
