Actually, your assumption in your first poster is 100% wrong.
It does use the energy of the star, but NOT for the weapon itself - it fires using the energy of "Dark Energy", as we see in the dialog between Ackbar and Finn (quoting from Novelization):
“As near as I understand it,” Finn continued, “enormous arrays of specially designed collectors use the power of a sun to attract and send dark energy to a containment unit at the core of the planet, where it is held and built up inside that containment unit until the weapon is ready to fire.”
“Impossible,” Ackbar insisted. “Although we know there is more dark energy in the universe than anything else, and that it exists everywhere around us, it is so diffuse that it can barely be detected. Let alone concentrated.”
Finn persisted, despite the discomfort he felt at disagreeing with someone of Ackbar’s rank and experience. “It can be, and it is,” he responded with certainty.
Statura, at least, seemed ready to believe. “If the engineering could be worked out,” he observed, “one would have access to an almost literally infinite source of energy.”
So, this is entirely new "physics", having nothing to do with the star, or E=MC2
Also, to address your further questions:
I'm curious whether the star wars canon ever mentions the ability of anyone to create/weaponize singularities or to otherwise manipulate gravity at a solar-system level scale
Not in Disney canon. But EU/Legends canon has Celestials who were indeed doing engineering on Galactic scale:
The Celestials were known to have created a number of projects that took place near Corellia, Kessel, and other sites in the eastern galactic disc. Notable achievements of this race included assembling the Corellian system, the Vultar system, the Hapes Cluster, the Kathol Rift, and the Maw black-hole cluster.1 (all sources and citations are EU/Legends level, on Wookiepedia) ...
Thus it was generally accepted by researchers that the Corellian system been assembled at some point from pre-existing planets that were brought together by a combination of buried planetary repulsors and hyperspace tractor engines like Centerpoint Station or the Cosmic Turbine.1
As a matter of fact, I wouldn't be surprised if the latter (hyperspace tractor engines) and the weaponized use of Centerpoint Station from Corellian Trilogy were inspiration for Starkiller Base in Episode VII - Disney openly stated that they will
steal reuse and Legends concepts they like in New Canon.