If it were variations in the sun's behaviour that caused a whole lot of global warming during the PETM....how would we find physical evidence of that now?

Furthermore, could it be possible that an increase in solar energy during the PETM would have itself caused an increase in atmospheric C02? Quite possbily, I should think, because if the sun heats the surface of the Earth more than usual then you will have a greater incidence of drought over wide areas of geography, and that will cause a greater incidence of forest fires and brush fires to occur. More forest fires and brush fires means a lot more smoke and C02 going into the atmosphere. (Volcanic activity can also put a lot of C02 and various pollutants into the atmosphere...but that can cause global cooling for awhile by blocking sunlight.)

It's quite possible that the changing activity of the sun has been the major player in Earth's past heating and cooling phases...not greenhouse gases. An increase of decreans in greenhouse gases may itself be partially a biproduct of changing solar behaviour.

If so, well, we can't do anything about the sun. All we can do is adopt various measures to better cope with the effects of climate change.

There will always be climate change in one direction or another. What humanity needs to do is get better at anticipating a major climate change before it happens and taking measure to protect people from the worst effects of it. Preventive medicine, in other words.