What are the theoretical / mathematical problems in discarding negative solutions of Dirac equation?
The problem is that Dirac equation can't be written as two equations, where one would only refer to positive-energy components, and the other to the negative ones. E.g. $\partial_t\psi_1$ component depends on $\psi_3$ and $\psi_4$ in the equation. The result is that, if you find the general solution of the equation, you'll see that for nonzero momenta the components are intermixed, and you only get pure positive/negative solutions for particle at rest (see this post for explicit solutions).
All this makes rejection of negative energy solutions not only "physically unwanted", but mathematically impossible.