What fallacy is being committed here?

I think this is simply a false premise.

Evolutionists don't claim evolution is true because it is possible; they claim evolution is true because using the scientific method, there is substantial evidence that it is true.

How did life start is a recurring question. The mystics of whatever denomination claim some god or other did it. The scientists, though, I thought had well settled the question by Stanley L. Miller and Harold C. Urey in the 1950s. Of course, a few scientists contradicted that the gases used by Miller and Urey are not as abundant as shown in the experiment, but I found that to be a weak criticism. 

A premise in the question here is: “Despite the mathematical improbability ...” That is a false, or at the very least an unsubstantiated, premise. Before one can hold that as a premise, one must establish that there is a mathematical improbability — a point I think Sagan and Dawkins would argue is untrue. 

Despite the mathematical improbability of such a thing happening...

Creationists fail to compare this to the "mathematical improbability" of magic from a god.

evolutionists fallaciously cling to the position

This is most likely a strawman fallacy because the implication is that evolution is being accepted "fallaciously" or unreasonable, without knowing why evolution is accepted—assuming we are talking about evolution here and not abiogenesis.

The creationist is also conflating abiogenesis with evolution, which is common. Again, might be a strawman because they are assuming the evolutionist also subscribes to abiogenesis—possibly with the same conviction they accept evolution.

While speaking of math... we have billions of examples of things that happened naturally, and exactly zero examples of things that happened supernaturally (i.e., by magic), so preferring abiogenesis to magic is more than reasonable.

fallaciously cling to the position that if a thing is POSSIBLE to have happened that it DID and DOES happen

Again, almost certainly a strawman. While there may be some people out there that accept abiogenesis or evolution primarily based on the appeal to possibility , I doubt this at all common.