Staging the coup would not be easy. But the problem would be the Nazi Army in France - they might decide to move some tanks to help out Franco. This might be part of the plan - keep the tanks busy, while the allies invade Normandy.
The OP says, after WWII. Messing around with Spain during WWII is a whole different topic. On the one hand, there is more reason for the Allies to be suspicious of Franco as his fundamental allegiance is to fascism and very strongly against the Soviet Union. However, one of the paradoxes of fascism is that hypernationalism is a key element of its ideological stew, which would seem on the face of it to pose an inherent barrier against any sort of internationalist "Axis" of fascists. In fact the SS found interesting ways to square that circle, and the brutal fact of the matter is that formation of an Axis is about raw power, subordinating one small nation's fascists to subservience to the big nation's fascists, and peer powers who are in a position to either cooperate or not, such as Italy with respect to Germany, negotiate partnerships based on perceived mutual benefits.
Spain was in the position of being a small, weak fascist state that however had some strategic negotiating power with respect to Hitler's on paper overwhelming strength. Franco for his part demonstrated, for all his faults and failings, a determined Spanish nationalism prioritizing himself and Spain first and second, and understood shrewdly enough that mere submission to Hitler would benefit neither. Had he somehow placed Spain at Hitler's mercy by inviting substantial German military force in, as would have been logistically possible after the Fall of France (though it is not clear to me any foolish decision on his part to do that would not be resisted by others below him in the Spanish Nationalist hierarchy) Franco would become quite dispensable and his life dependent on delivering perfect satisfaction to Hitler personally. And Spain would suffer terribly. She would lose her overseas colonies forever, and be deprived of overseas trade Spain will suffer badly without by RN mastery of the sea. That's
before the Allies start considering whether they need to intervene militarily. No doubt if Franco handed the keys to Spain over to Hitler, Hitler would seek to take or at least neutralize Gibraltar, and this raises the priority of the Allies doing something to stop it, turning Spain into a battlefield again.
With Hitler's huge and recently victorious army right there on the Pyrenees, not to mention the debt of gratitude Franco owes the Axis powers for their help in the Civil War, Franco can hardly consider joining the Allies either, even if he could swallow the idea of working on the same side as the Soviets.
What he did OTL was smart, and the only smart move he could make--he temporized with Hitler, negotiated, haggled, and stalled. Spain's technical neutrality had some serious costs for the Allies but it also brought benefits. On the whole, the way Franco played it, it was never worth either side's while to upset the applecart and intervene for their greater benefit, because the cost of trying for that would be considerable--until the situation evolved where the benefits would cease to matter. Once the Allies took Southern France back, Franco would be dead in the water if the Allies had any reason to prioritize cleaning his clock. But by that same token, Nationalist Spain was no longer any kind of asset for the collapsing Reich, not as a sub base, not as a source of resources--it was and remained a place where Reich spies and later refugee collaborators and German Nazis could try and hide out from justice, but Franco largely caved in to pressure to extradite these fugitives. I think the rule was, the Spanish police were just amazingly oblivious, being unable it would seem to recognize infamous Nazis or collaborators from photos, but if some foreign agents such as British intelligence or OSS/CIA or Mossad were to haul them into the station, into the lockup they'd go and extradition to Nuremberg trials or Israel or Denmark or wherever would proceed from there, by proper due process.
But this is getting back into the postwar period. During the war, Franco played both sides off against each other so that it was always more worthwhile to leave Spain alone and focus the war effort elsewhere.
I've fantasized myself about the ATL "benefits" the Allies might have reaped if Spain were somehow on the allied side AND it were possible to hold the line pretty much at the official border with France when the Allies were weak, and thus have a fixed broad beachhead already established to pour into Southern France from at whatever moment Allied command deems suitable. That's a nice daydream (plausible only in the context of first of all Republican victory in the Civil War and then French forces fleeing collapse of the Third Republic being adequate to hold the line of the Pyrenees, and possibly that is a pipe dream too) but let the Reich find a break in the armor anywhere and enter Spain, and once again Spain is a battlefield, and the Allies must conquer their way to a Pyrenees line the Axis holds and fight their way over its top. Not so great then really! Franco of course, even if he had no animosity against the USSR, would surely want to avoid this just to spare Spain more such turmoil.
For either side to upset the balance in Nationalist Spain, Franco has to do something stupid.