Would there being no Soviet Union affect Spanish politics, if there wasn't an existing Communist bogeyman to see as an invisible hand in the republican government?
That´s easy. The fascists started the coup in the African colonies 24 hours before the coup in the Spanish mainland began. Trade union leaders and so on asked the government for weapons immediately. They wanted to blockade the barracks of the Spanish army, to discourage or trap and rebellious units. Wherever they did that, the troops either remained loyal or were quickly defeated.
That was option No.1, here is No.2:
The governments of other democratic countries permit the sale of arms to the democratically elected government of Spain while taking steps to prevent or at least reduce arms shipments to the fascists.
And last but not least No.3:
France, faced with the prospect of being surrounded by fascist dictatorships answers to a call form the spansih government and deploys a two or three regular divisions –perhabs the FL or colonial units- to help put down the coup.
Given the lack of guts and back the governments of France and not-so Great Britain showed, the best way to handle thing is No.1.
I think I've got a POD that prevents the Spanish Civil War. In 1933, instead of choosing Lerroux to be vice-president, invites Gil-Robles. According to my sources, if he was VP, he would have reacted to the turmoil leading up to the Civil War a lot better than Lerrous did OTL. That's something to work with, at least.
A quick coup would not have worked either. For Franco was in chrage of the most experinced faction of the military at the time, and he was given somewhat de facto control of the operations. Much of his planning involved a long war to get rid of as much Republican forces, and sympathy as possible.
Say what one likes but Franco had an amazing ability to play sides against each other, and see the bigger picture.
But the initial plan was to seize power ASAP. The longer war may have suited Franco, but it wasn't the intention. However a quick successful coup probably does not result in Franco as leader, but rather some junta led by Sanjurjo or Mola.
A Falange government would be interesting, but hard to achieve with such strong anarchist and socialist opposition.
Ultimately though, the easiest way to avoid the civil war is to take an early POD, and have some reforms instituted by Rivera or by a subsequent regime that defuse some of the tensions. Limited land reform or taxation reform could do it.
Also for the quick action to work you need so much control that the nationalists did not have. They expected full naval support, but while the officers where nationalists the crews supported the Republic. Without the navy the 30,000 Army of Africa cannot get to the mainland in proper order. Conversly the Government needs control that it just does not have either, for in the same amount of time that the Nationalists rose up, anarchists and communists did as well in places like Asturias and Catalonia