About the possibilities of Spanish limited intervention in WWII…..IMHO there are no powerful motivations for TTL Spain to join the Axis. Firstly because in spite of some vague ideological affiliation with Axis powers, TTL Spain , as OTL, would remain dependent on oil from Allied side, and the Spanish air would be still too vulnerable to the RAF in case of eventual entry on Axis side (a factor which IMHO cannot be completely butterflied away despite a stronger military build-up ITTL ). Itis true that irá population would be less exposed to famines if SCW hasn’t took place, hence the country would be less dependent on food shipments….
But, could the Axis offer anything interesting for TTL Spain to put at risk her neutrality? IOTL Franco demanded the cession of French N Africa for joining the Axis, but his demands were too ridiculous considering his poor military capacity, therefore Hitler got fed up of Franco very early. Axis preferred a neutral-friendly Spain, especially after seeing how a minor partner like the Italians had screwed it in Greece….
Strategic importance of Spain was determined by her position in W Mediterranean. With the capital exception of Gibraltar, Spanish presence on both sides of the Straits made her desirable to both warring sides, but in the end both of them were content with Spanish neutrality. The most critical phase for the maintenance of the status quo was the implementation of Operation Torch, where a pro-Axis Spanish Moroccan Army could have ruined Allied plans.
Besides this, and ruling out the TLs regarding Allied landing in Spain (very impractical), the option of the Spaniards joining the Allied effort isn't out of the question…..but from a very comfortable position, that is when the Axis defeat seems inevitable.
One option could be just declaring war on the Axis just like Turkey did, without firing a shot, when the war is almost over and far away from Spain….
Another one could be
Operation Dragoon, the seaborne invasion of S France. Maybe Spain, after witnessing the result of the Normandy landings decides to give naval support to the operation.