Momentum and energy are referential dependent. You must choose one to describe momentum and energy, and keep the same before and after the collision. The simplest referential frame is the own referential of one of the balls before they hit each other, because in this referential, one ball speed is null, meaning that its momentum and kinetic energy are null. So for instance, we can always consider the small ball at rest before it is hit, question of choice of the referential frame. If you introduce a third referential frame in which the small ball move at 90°, as in your previous picture, it is of no help. We don't need it, it's a useless complication. From the viewpoint of the small ball, the big ball will hit it in the same way as if it had no speed. The small ball can be seen with any speed, just a question of viewpoint of an observer moving in respect to it: this will not change the slightest thing in the collision. Only the speed vector of one ball relative to the other at the instant of the hit is relevant. Your mistake comes from mixing different referential frames when you analyse the problem before and after the balls collide. Neither the momentum nor the energy are conserved from one referential frame to another.
It is better to go straight to an example. Using the website i provided. Let's say we have a small ball 1kg moving at +1m/s hitting a big ball 100kg at rest. The small ball will bounce back at -.980m/s and the big ball will move at .0198m/s. The momentum of the big ball is 1.9 and the momentum of the little ball is .98. Now we just use the big ball to hit the side of the little ball moving at .980m/s . It is the same as the big ball hitting a small ball at rest by ignoring the moving part (we hit it at 90 degrees). The small ball is now moving at .98m/s with a side speed of .039 m/s . The big ball is now .0194m/s The new speed of the small ball is now sqr(.98^2 + .039^2) = .9805 m/s . There is a slight increase in speed. The big ball still have speed to collide again at 90 degrees (yes we change frame again). The small ball again gain a little speed. After the big ball exhaust all of its speed. The small ball speed is sqr(.98^2 + 1.96^2) = 2.19 m/s I do constantly change frame of collision. Does that means it takes energy to do it? Yes and no.
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