Here the opposite of this question has been answered: Looking for a proof relativistic momentum is conserved Using first principals
But I want to know if the formula $$p^\mu=m_0\frac{ds^\mu}{d\tau}$$ can be derived from assuming four-momentum is conserved regardless of reference frame.
But I want to know if the formula $$p^\mu=m_0\frac{ds^\mu}{d\tau}$$ can be derived from assuming four-momentum is conserved regardless of reference frame.
That is too little to go with - statement like "four-momentum is conserved in all frames" does not say anything specific about what four-momentum is, not even that it should be connected to particles, or their mass, velocity or their trajectory or other things. You have to assume more things. Maybe if four-momentum is assumed to be a function of mass and four-velocity, then assumption of conservation in all frames can imply something about that function, but without those assumptions, there is nothing to cook with.
The more you assume the more you can derive.
Using only the assumption that it is conserved is insufficient. There are many conserved quantities, so that single assumption would not permit you to specifically find the four momentum.
However, if we start with the metric $ds^2=-dt^2+dx^2+dy^2+dz^2$ then we get the Lagrangian $L=- \frac{m}{2}(-\dot t^2 + \dot x^2 + \dot y^2 +\dot z^2)$. We see immediately that this Lagrangian is symmetric with respect to the coordinates. So we can use Noether’s theorem to derive the specific conserved quantity associated with this symmetry. That quantity is the four momentum.
