In 1975, and until the coup attempt in 1981 where Juan Carlos showed the entire world and Spain's political class that he was a true believer in democracy, the King was seen in the beginning as nothing less than Franco's puppet, the true continuation of the dictatorship. Maybe a harder left comes in power earlier, before Juan Carlos can claim himself as the true embodiment of Spanish nation and democracy; or, if we follow the Fear, Loathing and Gumbo scenario where the Carnation Revolution leads to a communist Portugal and gives shudders to Spanish generals, where the democratic process is stopped, and so the establishment of democracy would overthrow the monarchy as well.
Having Juan Carlos siding with the coupsters in 1981 is impossible to imagine, unless you change the King's behaviour, or if the coupsters are more powerful than the nostalgic rabble they were. Maybe if you have a different successor to Franco, more likely to get puppetized by the Francoists, such as Count Juan of Barcelona, Juan Carlos' father, or the latter's younger brother Alfonso, who died in a freak accident when he was a child.