I don't think so, at least not in a Soviet style communism. If the government has won by January 1937, it means that it has been strong enough to beat the coup or the rebels failed strepitously. Either way, outside influence has been minimal, which means the communist are a marginal group (due to a heavy radicalized Spanish Socialist Workers Party, true) and more importantly, anarchosyndicalists haven't been beaten down by their supposed allies and are in fact in control of Catalonia, at least half of Aragon and probably huge swaths of Andalucía and Extremadura. The victorious government will have to stop the anarchists, who have begun their revolution (they had, in fact, as soon as february 16) and will not yield, and that can be a thorn in the Popular Front that sets the coalition apart (see the events or massacre of Casas Viejas '33), initiating yet another civil conflict in Spain.
Don't forget there is also the POUM and various Trotskyist aligned groups. Assuming it gets involved in WW2 and I can imagine that it would since it would be too much of a thorn to Hitler. Since Sure the Republic needs time to rebuild that doesn't stop it from being aligned with the Allies and providing them with cover and support against Hitler. Which obviously means the Nazis would at some point seek to neutralize the threat. Perhaps Portugal joins the axis in this scenario and Spain is invaded by Italy, Portugal, and Germany?
What I would be curious is to see how a POUM/CNT-FAI influenced Spain would play in the Cold War assuming that everything else roughly goes about the same as in OTL. By influenced I mean that it would be similar to your idea but perhpas with the POUM/Anarcho-Syndicalists ideas winning out in the Government, but still a democratic spain in contrast to the communism of the USSR.