We can break the 5500K temperature limit with passive elements and some clever engineering.
Also, a solar pumped laser needs cooling, just like a heat engine! Just from first principles, you can’t have a gadget at thermal equilibrium that magically produces an excess of light at its output frequency — this would violate the second law of thermodynamics.
And if you imagine a model three-level laser, exposing all three transitions to bright light will pump the nonlasing intermediate transition (overfilling level 2, thus reducing the population inversion in a laser lasing on the 3 -> 2 transition, or pumping useful level 2 atoms back up to level 3 in a laser lasing on 2 -> 1) and will actively un-pump the lasing transition in any variety of laser (because light in the wrong mode will cause a population-inverted transition to emit into that same wrong mode by simulated emission).