kinematics - Hypothetical special relativity with mass conservation

I thought of this question while studying inelastic collisions in special relativity, where kinetic energy is converted into mass-energy.

I was wondering if it's possible to formalize a version of special relativity where mass conservation still holds. Basically I'm imagining that, in this hypothetical theory, the loss of kinetic energy in an inelastic collision would be accounted for in the same way as in classical kinematics: By assuming that the kinetic energy associated with center-of-mass-motion is converted into kinetic energy associated with disorganized, relative motion about the center of mass - that is, thermal energy.

Of course, this theory would be empirically wrong. But here is my question: Would it be wrong simply because Nature "decided" not to do things that way, or is there a compelling theoretical reason why we ought to doubt it? For example, would such a theory contradict Einstein's postulates of special relativity in some way? Or, perhaps, would it violate certain symmetries?

Thanks.

Edit: To make my question more concrete, here's an example of some calculations in this hypothetical theory, per Ismasou's request:

Consider a collision between two objects, of masses $m_1$ and $m_2$. Suppose these two masses "stick together" upon collision.

Given mass conservation, the total mass of the resulting object is just $m_1 + m_2$. Now, let's restrict our attention to the inertial frame where $m_2$ is at rest, and $m_1$ is moving at some initial speed $v_i$ (assume this is a 1-dimensional problem). What is the final speed $v_f$ of the composite mass?

Well, supposing that relativistic 3-momentum conservation still holds, we know that the initial and final relativistic momentum $p$ is given by:

$$p=\frac{1}{\sqrt{1-v_i^2/c^2}} m_1 v_i=\frac{1}{\sqrt{1-v_f^2/c^2}} (m_1+m_2) v_f$$

If my calculations are right, solving this equation for $v_f$ yields:

$$v_f = \frac{1}{\sqrt{1-\left[1-\left(\frac{m_1}{m_1+m_2}\right)^2 \right]\frac{v_i^2}{c^2}}} \frac{m_1}{m_1+m_2} v_i$$

Notice that in the limit of low speeds, the square root factor approaches 1, and we recover the result obtained in a classical inelastic collision problem.

Now, here, it is not obvious that relativistic energy $E=\gamma m c^2$ is conserved - it seems like we've had to give up energy conservation to maintain mass conservation. However, in this theory, relativistic energy might be conserved in the same way that kinetic energy is conserved in a classical inelastic collision: By postulating that the kinetic energy which moved $m_1$ forward has simply been dispersed into the many disorganized motions of the composite mass about its center of mass (thermal energy).