Are "constraint trees" in phylogenetic analysis cheating? How can I correctly use them?
I understand where you are coming from when you say that it feels like cheating. However, there are a coupled of things to keep in mind. First let's consider why your tree topology might not reflect concordance with intuition/species trees:

1) Presence of artefacts such as long-branch attraction

2) Lack of sufficient phylogenetic signal/presence of non-phylogenetic signal (due to, for example, convergent evolution in the sequences)

3)Horizontal gene transfer (unlikely in animals).

4) Incomplete lineage sorting/deep coalescence (hemiplasy). 

Only in the case of 3 and 4 does your tree reflect true relationships among taxa that genuinely differ from the species tree. Cases 1 and 2 are artefactual. In cases 3 and 4, it would not be appropriate to use a constraint tree to get a species-tree concordant topology because the resultant topology would not reflect the 'true' relationships among taxa (i.e. would be 'cheating'). In cases 1 and 2, I would argue that it is appropriate to use a constraint tree to get the species-tree concordant topology (depending of course on what you want to use the tree for) because you know that the your current topology is species-tree discordant because of artefacts and is therefore not reflective of the 'true' relationships among taxa. Of course, you are unlikely to know a priori what factors (1-4) are causing your bad topology (though you can do some tests for long-branch attraction for example -- see Cladistics 21 (2005) 163–193).

What you can do is test whether the log-likelihood of your non-constrained tree and your constrained tree differ significantly. You can use tree topology tests such as the  Kishino and Hasegawa (KH) test. I don't know about RaxML, but IQ-TREE has excellent topology testing: http://www.iqtree.org/doc/Advanced-Tutorial. If your constrained topology is significantly different (statistically) from the ML topology, then you probably should not use the constrained tree. I am by no means proficient in phylogenetics concepts so I would take my advice with a grain of salt, hopefully some of the other members can illuminate the issue a bit further.