Dust spots that get more visible and better defined as you stop down are most likely from dust on the sensor assembly—more precisely, on the filter stack in front of the sensor—not from dust inside the lens.

Dust in a lens usually doesn’t show up as distinct spots in images because the light effectively diffracts around small obstructions. Very heavy internal dust can reduce contrast or cause other image issues, but it typically won’t create the classic sharp dust specks that become more obvious at smaller apertures.

So based on your symptoms, sensor-side dust is the likely cause. As a practical workaround, using an ND filter to let you shoot at wider apertures is a reasonable low-cost option if you want to postpone service.

For repair, either issue would generally require disassembly on a fixed-lens camera like the LX100, but the evidence here points to sensor/filter-stack cleaning rather than lens dust.