Which would be safer: removing hydrogen or adding hydrogen to our sun? If we want to extend the lifespan of our habitable zone safely, would it be safer to institute a yearly 'tax' on our suns outer layer of hydrogen, or a yearly 'subsidy' to our suns outer layer of hydrogen?

(the feasibility of said tax/subsidy is beyond the scope of this question)

Another way to ask this question:
Assuming the core of the sun remains untouched,
but its outer layer of hydrogen were reduced by half,
would the suns main sequence remain intact for a longer time?

To avoid ending up with a white dwarf and its associated bad weather, what would be the ideal point to reverse the tax and re-subsidize?
My kneejerk reaction is that your only option is to remove a chunk of mass from the outer part of the Sun.

The Sun will respond (on a Kelvin-Helmholtz timescale), by contracting and becoming less luminous because the core temperature is lower in a less massive star. This will extend its main sequence lifetime, because only the central parts of the star are involved in fuelling the nuclear burning. The core is separated from the well-mixed outer convective envelope by a stable radiative zone.

Since the default solar evolution involves it getting slowly more luminous with time, it is possible that mass could be extracted at just the right rate to keep the Sun at constant luminosity. This sounds like an interesting calculation to do.

The alternative of adding "fuel" in the form of hydrogen won't work. The Sun's luminosity will increase because of its increased mass and increased central temperature. However, because the new fuel cannot be mixed into the core (only the outer part is mixed by convection), then the main sequence lifetime would be reduced.