This goes right to the core of unitary theory -- that there is only one true theory of morality. But I must admit I'm dumbfounded at how any one particular theory of morality could be "the one true one", except in so far as someone personally chooses that theory over others based on preferences and desires.

Think of morality, not as solipsistically fulfilling personal desires, but as a means of resolving conflicts between desires (within groups). Why would it then be impossible for it there to be an optimal way (amongst groups) of doing so?

but I must admit I'm dumbfounded at how any one particular theory of morality could be "the one true one", except in so far as someone personally chooses that theory over others based on preferences and desires.

If N people choose differing One True Moralities, then there are N True Moralities, so that doesn't work at all.

This gets me a bit more background knowledge, but what is the ontology of morality?

Thinking of morality as conflict resolution, it is then something like group desicion theory or economics...those things do not require a special ontological realm.

How does morality get it's ability to be rationally binding?

One answer is that it gets it the same way maths does, If an agent can that there is a good arguemnt for X AND it has an desire to believe rationally demonstrable things in general THEN it will be bound by what can be proven to it, or what ic an prove to itself. To sidestep this argument, you have to assume not ONLY that rationality is always in the service of desires, but ALSO that desires cannot possibly include desires to be maximally rational, to believe truth for its own sake, etc (in contravention of the Orthogonality Thesis!)

If the very definition of "rationality" includes being moral, is that mere wordplay?

Who siays it does? There are ways of inferring conclusions other than pulling them straight out of definitions.

Why should we accept this definition of rationality and not a different one?

You don't get a free choice. Why would any old definition count as a definition of morality? I could define "dog" as "middle C played ona n oboe"...but I would be talking nonsense.