This paper claims that FastTree is almost as accurate as RAxML, while being much faster. You just have to be careful, however, that the support values output by FastTree are not bootstrap values, they are based on the Shimodaira-Hasegawa test. (Also, see this comment for the case you have very short branch lengths). [update: However, according to the recent comparison paper mentioned below FastTree performed quite poorly in comparison to RAxML or IQ-tree.]
From what I understand, you should use ExaML only if your data is too large to be handled by RAxML in a single node. ExaML should perform like RAxML but with some parallelization overhead. For all effects I treat them as the same. I don't know of relevant advantages of phyML over RAxML (for me, it's easier to use but I am very used to phyML).
I am not familiar with IQ-tree, but its authors claim that even given the same time as RAxML or phyML, IQ-tree already finds better likelihoods more often than not (although by default it takes a bit longer to converge). A recent comparison between all these programs favoured IQ-TREE for both single-gene and concatenation analyses (with RAxML very close). It may also estimate branch support through a SH-like test only, but I'm not sure. [update: IQ-tree offers 3 measures of support, standard bootstrap, aLRT, and ultrafast bootstrap. See OP's comment below for details.]
However, since you have few missing data, you might also want to try a single-locus tree inference followed by gene tree clustering (using treescape or treeCL) to see how spread your data is, or to see the effect of removal of outliers, or to use ideas similar to statistical binning.