@Dave
...also, as well as being inefficient and only working at high intensities, nonlinear optics which produce sum-frequency generation only work at specific wavelengths.
Mercury vapour lamps work by exciting the vapour, by passing an electric current through it. This produces a series of specific wavelengths, known as an emission spectrum. This is because the atoms in the mercury vapour are separated, as a gas. It is specific fequencies in the ultraviolet that excite the phosphor to produce reddish light, to counter the otherwise blueish light the mercury produces in the visible spectrum.
In a metal element, such as the tungsten in an incandescent bulb, the atoms are together in a metallic state. For several boring, quantum physics related reasons, exciting such a material produces a band of emissions, rather than specific frequencies. This is to do with the way electrons can move in a metal.
The upshot of this, is that to do any frequency doubling, or other sum-frequency operations would require a number of materials, each tuned to a different frequency, or set of frequencies, to absorb and reemit the light at a range of shorter, visible wavelengths. The R&D and manufacture of such composite materials would be prohibitively expensive, so the costs would outweigh any benefits by several orders of magnitude.