Do we know anything about the nature of Earth's core that hasn't come from magnetic or seismic measurements?
Short answer: yes, most of our knowledge is not from seismics and geomagnetism, but from experiment and measurements, deductions from geochemistry and -physics, material science and physics in general (besides the already mentioned measurements of earth's gravity, but the variations are rather connected to mantle convection, water storage, etc. than the core). But all in all those weigh much more than "painstaking" seismics and geomagnetism.

One experiment to replicate pressure/temperature (p/t) conditions is the diamond anvil cell and its cascaded version in combination with laser heating of the probe, to obtain data about mineral and cristalline phases and transitions under conditions in a planet's core.


Wikipedia: Diamond Anvil Cell
Teaching Mineralogy; Topical Resources The Diamond Anvil Cell (DAC)


And some random links on the outcome of the matter:


Reviews of Geophyics: High-Pressure Experiments and the Phase Diagram of Lower Mantle and Core Materials
Nature Communications: Implementation of micro-ball nanodiamond anvils for high-pressure studies above 6 Mbar
Earth and Planetary Science Letters: The Earth's core composition from high pressure density measurements of liquid iron alloys
Geophysical Research Letters: Iron-Nickel alloy in the Earth's core


Laser pulses is a developing technique to reach very high p/t conditions, higher than diamond anvil cells can, but they do not last that long


Science Advances: Crystal structure and equation of state of Fe-Si alloys at super-Earth core conditions