I think anti matter fits your use-case well.
Benefits of Antimatter
- Very simple to moderate the amount of draw, low minimum power threshold
Your energy extraction is directly proportional to the amount of mass you input into the system. Many other power systems like Fusion, Fission, or Hawking Radiation all have minimum output thresholds or require a minimum amount of energy to self-sustain.
If you don't want power out of an antimatter battery, you just don't input mass - so it has a minimum energy generation of 0, in principle.
Of course, containment fields etc. will still have some kind of 'vampire draw', but as a percentage of overall output it should be much lower than competing alternatives.
- Extremely high energy density
As a fuel source, it's nigh impossible to beat antimatter. If you want mass -> energy conversion of any kind, this would be how to get it done, as the energy contained by your reaction mass is equal to MC^2. It's difficult to quantify the energy demands of your space ship, but a gram of antimatter could power LA for about 2 years, give or take.
- Mechanical simplicity
To trigger a reaction that generates energy, just touch the anti-hydrogen. Although containment fields are non-trivial to engineer, this is actually pretty simple - it's just a magnetic cage in a vacuum chamber. It could be completely solid-state, in principle. The hard part about generating power from anti matter is just getting antimatter in the first place, but your magic aliens have obviously solved that problem already.
- Completely self-contained
An antimatter power generation system does not require any external force operating on it to produce power - it doesn't need to get hit by photons or neutrinos, experience significant gravitational interactions, etc. This means it can continue to operate anywhere, under any conditions, as long as the reactive mass doesn't run out. As a side benefit, because it's totally self-contained, it can also be nigh undetectable (with adequate shielding). A power source that reacts to external inputs such as a gravity wave generator or even Neutrino voltaic panels would leave a trace of some kind as it stole energy from it's environment, whereas this system would leave only waste heat.
Drawbacks of Antimatter
There are, of course, some problems with this solution that may render it unsuitable for your story.
- Minimum mass
You need an amount of reactant mass equal to, well, E/C^2. So depending on total 'E' expenditures for 130 millenia, that could be quite substantial.
Your power source is a really great battery, but it's just a battery. It can run dry. Eventually.
- Requires maintenance
This is not a drawback unique to antimatter, as it applies to all possible power generation options, but it is something to think about.
You can't escape it.
Everything, and I do mean everything, degrades to some extent over time.
You can't run a constant current through a wire and expect it to stay pristine - you will need something to replace power conduits, electromagnets, etc. Even assuming exotic materials that mankind has never seen, 130,000 years is simply too long for something to exist without experiencing wear and tear.
Your ship will require some system or robotic staff that is capable, in principle, of replacing and retrofitting any individual component of the ship. Given advanced future tech like molecular printing and/or atomic forges, it's not inconceivable that a perfect (or near enough perfect over the course of this timeframe) repair crew could exist - but you'd need to have one, and it'd need to be active periodically to affect repairs.