Would a laser with four possible energy levels be better than three?

When you have a three-level system, the laser transition is between the ground and first excited levels (see figure). In this scheme, it is rather challenging to get population inversion because all atoms tend to stay on the lowest level.

With a four-level scheme, you have an extra level so that the laser transition does not end in a ground state. Thus, if the bottom level gets depopulated faster than the top level in the laser transition the population inversion will be guaranteed independent of how fast or efficiently you pump the system. Many lasers, such as Nd:YAG or HeNe laser to name a few, actually use a four-level scheme exactly for this reason.

Three- and four-level laser setup.

If you have a fast non-radiative transition to the ground state from the bottom laser level, that would help with population inversion because you're emptying the bottom laser level.