Yes, the best chance is Juancar making the bad choice the 23-F (1981).
If not, that depends on the degree of nastyness that you want in the recent spanish history.
Option one. ETA kills Juancar during his holidays in Majorca before 1985. Improbable, since, as far as we know, the plans to that were made in the 90's
Option two. The PCE, or better said, Carrillo, don't accept the Monarchy. There is some posibility of contagion to other democratic pàrties in the left. A more actively republican left means a more actively monarchic right and a worse politial climate in he late 70's early 80's, and probably more violence than IOTL. But maybe in the end they manage to force a referendum about the issue that probably the republic would win.
Option three. NATO polemic becomes mad. The first democratic governments (UCD) pushed to the entry in the NATO against the will of the majority. The kig was for. The PSOE was, theoretically, against. But IOTL when Felipe González won the 1982 elction, they suddenly changed their view. On the subsequent referendum, the participation was lower than expected, because many socialist voters didn't go to vote. The yes get a marginal victory.
A option to undermine the monarchy could be the PSOE keeping its word and the king actively defending the yes, butterflying OTL honeymoon between the socialists and the monarchy.
Of course all the three are very impausible. When Carrillo coined the expresion "Juan Carlos the brief" he ignored an historical reality. The spanish Bourbons may have a lot of defaults. They are not very intelligent, they are not hardworkes, the are not very cultivated. But they have an innate capacity to make anything to keep the crown, thanks to their ideological inconsistence (or perhaps causing their ideological inconsistence). And Juan Carlos is not an exception.