The first stage does use an artificial Reissner–Nordström black hole, or other supermassive and robust "shaver" object. This is dropped towards the star and placed in an unstable orbit, near enough to create a workable Roche lobe inside the target star.
This allows bleeding mass off the star until it reaches the Roche vertex at the expense of both the star and the "shaver" object's angular momentum (the latter will have to be replenished using the same engines that moved it into orbit).
At the intersection of the two Roche lobes there is a saddle in the gravitational potential, and in that point the infalling mass can be diverted with comparatively little effort.
By carefully balancing the shaver's orbital parameters and using appropriate electromagnetic capture and accelerator fields, it ought to be possible to "bleed" mass off the target star and impart it enough kinetic energy to send it into a far orbit to form a "smoke ring". We will probably need a good number of "shepherd satellites" to keep the ring stable (in the picture, of Larry Niven's Smoke Ring, the main shepherd is the darker spot bottom right - Goldblatt's World).
At the same time it should be possible to bleed some of the excess thermal energy from the incoming mass to reduce the total energy expenditure (the gravitational binding energy of a star is something huge, and however we dismantle it, even with leaving the massive core in place, we will require an unholy amount of energy - the difference between the gravitational binding energy of the star and that of its remains plus the Smoke Ring).
At the same time, the mass loss will cool down the star, extending its life considerably, gradually enough not to trigger a core catastrophe. Ultimately, it might be possible to completely evacuate all fuel mass, leaving the burnt-out core alone to cool off.
In stars whose core is above the Schönberg–Chandrasekhar mass limit, the cooling off of the core (and actually the removal of too much of the fusing fuel) will trigger a core collapse, possibly accompanied by several "flashes" when the progressing collapse ignites higher-order fusion episodes in the outer envelope (helium, carbon etc.). This might present a risk to the integrity of the Smoke Ring.
Very large stars probably cannot be safely bled past a certain point; as soon as the fusion slows down, the star will begin to contract and heat up, comparatively rapidly burning through the carbon, oxygen and silicium stages. When the carbon process stops, there are less than one thousand years to a respectable supernova-like explosion. Unless somehow (more bleeding satellites?) a lot of mass is removed at great speed, easing the pressure and slowing down the collapse.