Thanks for all the ideas so far! I have not tried anything too drastic yet. I washed the exterior and inside windows this weekend, so I took a bit of time to use some Glass Plus(non-amonia for the tinted windows on another car) around the button, but I was not very liberal with it. The button getting stuck only improved slightly, if at all. I will try next with some Windex. I will also try compressed air. I am still wary about removing the faceplate because I think the buttons will not be exposed enough. If I used the button more often, I would consider getting to the faceplate via the TVandNAV2Go DIY.
Thankfully, the button still does what it is supposed to, so its just a matter getting the gunk out.
'07 E92 335i Sport