You're not crazy to do it on the secondary side; as long as your transformer only draws on load reactance (somewhat subject to temperature, spot-on 60Hz AC, what you do with VOMs and shunt resistors if anything,) you can simply draw current evenly enough to match your rather particular 1ppm spec and somehow raise a flag on whatever needs changing if a capacitor goes out of spec in the lifetime of your supply. Moreover you can skip the isolation transformer and pick an RC, LC or LRC conductance mode filter to feed your baby charger/supply (which does sufficient AC/DC isolation in its own components, in exchange for attention to voltage tolerances, leakage, etc.)
Your spec of 1ppm is kind of nuts; it sounds like a solar or other energy harvesting (The Hills Have Ls?) supply is needlessly being put into a cheap inverter that can only tolerate these expensive (in parts or engineering, or both) DC supplies, when it's trivial to make adequate inverter or a DC (even 24V) tap. Not to say it can't be awesome the way you initially called it, shorting a cap on the primary's not going to help reactance noise and guarantees the power loss which premises PFC equipment could have mitigated. Spec of <1ppm on the noise, drift and wvdc is different; and do you need that to stay true close at 0v, where you suddenly need two or more extra decades (after 6) of hardness to RFI? Are Fairchild and ADI design models, which are almost certainly overkill but permit you keen instrumentation, truly inadequate so far?