In the OTL Maximillian I was proclaimed Emperor of Mexico in 1864 and was executed in 1867. Is it possible that Max ends up in Chile as Emperor or King rather than in Mexico. What changes would need to occur in Chilean society for them to change their system of government to a constitutional monarchy? Alternatively, are there other candidates that might make be a better fit?
Long answer, is yes, but caveats and POD starting from at least the Independence movement, short answers is NO.
All the founding father were rather Republican and not liked the idea of them being king, but were really open to some form of noble lineage, but in the end the only proper nobles that existed in Chile, minor nobles and counts, were rather pro-spanish crown and after the war were stripped of their ranks,and Chile was declare an equal republic.
But the Conservative consituttion was rather unitary and the President was for all intent and purposes an elected king governing for a period of ten years It´s not impossible to just consider that is better to have a king, with the possibility from the parliament to depose the king and elect a new one, and No european Noble will ever accept that