Maybe it's hard to explain from a real world example like a rocket but if mass isn't converted to energy in that case, in the E=mc2 sense, where does the energy come from?
The first one is the action-reaction principle: fuel goes in one direction, rocket goes in the opposite one.
The second one is indeed the E=mc^2 equation: as the fuel burns, some of its mass (a negligible part, but still results in some decent amount of energy) is converted into energy and this energy can be used to propel the rocket. The rest of the mass is ejected, see first point.
So if you compare the mass of the rocket + fuel before the take-off and the mass of the rocket + burned fuel after the take-off, you will indeed see a delta, which comes from a partial conversion of mass into energy of the fuel during combustion.
Actually, as pointed out in an other comment, every reaction (or most of) leads to a decrease of the energy of the objects: an empty battery has less energy than a charged battery (yes I know I'm a genius for pointing that out). The difference of energy results, according to m = E/c², to a loss of mass. In the case of a battery, the delta is waaaay to small to be detected, but it exists.
E=mc² is not about nuclear reactions, it's about every reaction. A chemical reaction of combustion transforms some fuel molecules into new molecules. If you compute the mass of the products of the combustion and compare it to the mass of the fuel before burning, you will notice a sliiiiight difference (probably negligible, barely detectable). The disappearing mass has been converted to kinetic energy (and heat, but who cares).
If I heat 100 molecules of water ice to melting, the 100 molecules of liquid water aren't more massive. The energy went largely into kinetic energy, not changing the atomic number of the atoms. Ok, that's a phase change, not a chemical reaction. I've changed the momentum, not the mass. Are you saying this would be encompassed in the full form E^2=m^2 c^4 + p^2 c^2 ?
[1] https://www.huffingtonpost.com/quora/is-a-hot-object-heavier...
EDIT> Is there any way to derive this outside of GR?
EDIT2> I notice that you were careful to say "heavier", not more "massive". Does this point to the effect being part of the gravitational interaction specifically?