Good news first: the touch screen lamp is at fault, not the transmitter. Touch screen lamps aren't supposed to respond to RF in any form, whether the fundamental signal, spurious or harmonics!
Most likely, the lamp's proximity detector circuit isn't bomb-proof (the necessary components are often omitted), and is being activated by a strong electric field. A different make of lamp might resist better.
The interfering signal could be direct or indirect via the house wiring which can act as an efficient receive antenna and pick up energy from the near field by induction. The near field is a zone of high intensity close to the antenna, weakening rapidly out to about 2 wavelengths, but allowing strong inductive pick up in any wires within 0.159 of a wavelength. So, an antenna transmitting on 14.2MHz (21.1 metres), should ideally be at least 3.4 metres distant. Antennas are best positioned as far and high from nearby structures as practical.
Most likely, the house wiring picking up the near field and is feeding a whopping RF signal into the lamp, which can't cope. House wiring can also respond to the far field. In this case maximum pick up occurs when wire lengths are at quarter waves multiples. This is quite likely in house wiring runs: 5metres, 10metres, 15metres. Might be possible to fix the lamp by shortening or lengthening the lead, or – more practically – by fitting a choke near the lamp. This could be home-made by wrapping several turns of mains cable around a suitable toroid, or commercial items are available. Water and gas pipes, telephone cables and similar can also cause trouble.
The nature of the antenna may be relevant. Not familiar with the sigma, but an antenna matched at the base with a balun is almost certainly unbalanced, which can cause local pick-up. There's a high-risk that despite the balun the outside of the coaxial feeder will become part of the antenna and radiate all the way back to the transmitter. They're noisy on receive too. May be worth choking the down-lead near the top by winding 20 or 30 turns of coax around a 4 inch plastic drainpipe.
Spurious and harmonics are significantly reduced by an ATU even if it isn't tuned optimally. However, although ATUs can usually get a low SWR at many different settings, some settings are better at nailing harmonics than others. Depends on the circuit inside, but best rejection is usually the highest inductance and lowest capacitance combination that achieves an acceptable match. I don't think this problem is due to spurii or harmonics though.
Given the best antenna that can be managed must be close to the house, low power is a good way of reducing trouble, and data modes are popular because steady signals are less likely interfere than peaky intermittent ones like SSB. Balanced antenna cause fewer problems if one can be fitted in: it's hard to beat a 1/2 wave dipole or a centre fed tuned-line antenna performance wise too. End fed antenna are easier to accommodate but more likely to cause RFI.
Probably worth improving the earth. Earth rods aren't good for this purpose because RF doesn't penetrate far into the ground. Only the first 300mm of an earth rod is likely to be effective, so current flow is restricted by the surface area of the thin rod and the conductive quality of the top soil, which is probably very poor. Good radio earths are close to the surface and consist of multiple copper wires radiating out over a large area. A quarter wave of wire, sometimes called a counterpoise, (about 5 metres for the 20m band) laid close to the surface is often a better radio earth than an earth rod. (But note a counterpoise is useless as a safety earth!)
Faraday cage – good idea, but the shield needs to be complete and well-bonded. Windows can be meshed over, but doors need special attention.
I have a different problem: my home is on a corner plot and surrounded by telephone and power cables on poles. The noise level is so high it's hard to make sense of anything other than strong signals.
Word of warning – radios don't like swarf, so keep them well protected from mucky metal working!
Dave