My reasoning is that since gravity is the only source of negative energy in the universe (under the assumption that all the other forces eventually cancel out without any overall effect) is gravitational potential energy. Kinetic energy (coming from the expansion of space; it will be the same in all reference frames), on the other hand, is not (there is at least the rest mass energy to take into account). That would imply that |GPE|>KE, meaning the universe will eventually collapse in on itself. Is my reasoning right?
Conservation of energy doesn't apply to cosmology. General relativity doesn't have a globally conserved scalar mass-energy that can be defined in all spacetimes. There is no standard way to define the total energy of the universe (regardless of whether the universe is spatially finite or infinite). There is not even any standard way to define the total mass-energy of the observable universe. There is no standard way to say whether or not mass-energy is conserved during cosmological expansion. For more on this, see Total energy of the Universe .
There are speculative ideas to the effect that maybe there is some as yet unknown way to define the total energy, and that it would be zero. The motivation for these ideas is that then the universe could be considered as some kind of quantum fluctuation, as allowed by the Heisenberg energy-time uncertainty principle $\Delta E\Delta t\gtrsim h$. People have been batting this idea around for at least 30 years, and nobody has come up with any idea of how to make it meaningful.
Kinetic energy (coming from the expansion of space; it will be the same in all reference frames), on the other hand, is not (there is at least the rest mass energy to take into account). That would imply that |GPE|>KE, meaning the universe will eventually collapse in on itself. Is my reasoning right?
General relativity doesn't have this kind of split of energy into kinetic and potential, so the analogy with Newtonian gravity doesn't work. You cannot analyze cosmological expansion in terms of KE and PE.
At the time when people first proposed this idea of a zero-energy universe, our observational knowledge of cosmology was very crude, and dark energy hadn't been discovered. The universe seemed to be approximately spatially flat, and people understood this to be a theoretical problem. Part of the appeal of the idea was that there might be some way to use the zero total energy to explain flatness. Since then, the parameters of that whole discussion have been changed by dark energy, and most theorists these days are looking to some version of inflation as the solution to the flatness problem.
Continuing your Newtonian analogy, a flat universe without dark energy is analogous to a projectile moving at exactly escape velocity. It isn't analogous to the projectile falling back to earth, i.e., a Big Crunch.