What is the best way to actually make the Earth lose heat?
TL;DR: Stop pumping so much carbon dioxide into the atmosphere.
The easiest approach in terms of human effort would be to let the Earth warm up a tiny bit. Thanks to the Stefan-Boltzmann law, a tiny increase in the Earth's effective temperature will easily rectify the ~0.6 watts/meter2 imbalance in the Earth's energy budget. If the Earth was a perfect black body radiating at an effective temperature of 252 kelvins, all that would be needed would be a 0.17° Celsius increase in the effective temperature to make that energy imbalance disappear. Easy, right?
Thanks to feedbacks, that small increase in the Earth's effective temperature requires a significantly larger increase in the Earth's surface temperature, and that's ignoring the fact that humanity is currently making the energy imbalance worse. Temperatures are currently rising and will continue to rise for a while due to the large amount of CO2 humanity has already pumped into the atmosphere.
Continuing to pump ever increasing amounts of CO2 into the atmosphere (the "business as usual" scenario) will result in a 4° to 9° C increase in the Earth's surface temperature by the end of the century, with the uncertainty of the increase depending largely on the climate sensitivity to a doubling in CO2 levels. Even the bottom of that range represents a disaster. The only way to avoid this disaster is to stop pumping so much CO2 into the atmosphere.
Can we have thermoelectric (Seebeck) generators powering infrared spotlights pointing out into space or something?
You wouldn't want to use infrared spotlights. You'd want the radiation to be in the visible range. The atmosphere is opaque in the infrared.
Playing "what if", suppose every last bit of energy currently consumed by humanity (18 terawatts) is used to power massive air conditioners, with lasers venting the generated heat into space. Suppose we somehow manage to get 17 times as much cooling as energy consumed out of the system, resulting in 300 terawatts of cooling. Dividing that by the Earth's surface area results in 0.6 watts per square meter -- i.e., the current energy imbalance.
This is of course utterly unrealistic. We aren't plowing or harvesting our fields, manufacturing products, transporting goods, driving to and from work, heating or cooling our houses. To make this unrealistic "what-if" scenario worse, the hydrocarbons used to produce that energy are exacerbating the energy imbalance.
The best way to make the Earth lose heat is to stop pumping so much carbon dioxide into the atmosphere.