If the mass changes from m0 to m1, the velocity has to change by a factor of m0/m1 to conserve momentum. But to conserve kinetic energy speed has to change by a factor of sqrt(m0/m1).
Something else has to be involved to reconcile these conflicting velocity constraints.
Does the energy need to be conserved in the same form?
No. But it needs to be explained what other form it takes.
Mass?
How does that work out in the at-rest frame of reference, when the particle is not moving?
In our current physics everything is always vibrating or moving.