Well, the only thing that Chrome does is mask itself as Safari in order to give you access to office live. Whether or not Chrome (or office live) has the supportability is another. Basically, as you said is that it "pretends" but that is it. One thing to note is that any browser can any browser name but it is still up to how the original browser interprets the code. If that makes sense. So if I send Safari to Office Live I am still executing code as Chrome and not Safari.
Unfortunately, the only thing I could say to do is get with Google. I searched their help forums and you would not believe the amount of issues that Chrome has with Office Live. Wish I could have been more help but when it comes to browser interoperability there is little that we can suggest except to not use that browser. One of these days we will have one standard for the way browsers interpret all the code.