Joshua's post was extremely well done. I would like to add to his post this little tid bit. The kamiza, notice za at the end, is the same as the za in seiza. So the kamiza is the God's seat. In the old days and probably still today depending on the dojo, the Sensei would sit at the front on a little stool and conduct class from that location. The Sensei is the "God" of that dojo. Hence kamiza being used for the front of the dojo. Most people didn't speak Japanese when the arts came to the West so I'm sure you can understand the mixups with the terms.
"Kamiza" as the front of the dojo generally uses a different kanji - "kami" meaning "upper", so the "kamiza" is the upper seat while the lower level people sit in the lower seats.
I've trained at a number of dojo in both the US and Japan that had some kind of Shinto shrine setup - Aikikai hombu dojo used to have one too, but they took it down after the war.
So, what are you going to do? Could you imagine going to a dojo in Japan and saying - "I am not going to bow to the Kamiza because it is meaningless" ??? To Japanese, it has great meaning and they may not understand your point. And even if they did, they would probably not regard you with much respect. Personally, I am not religious at all, but I do show respect when required (when people say prayers at dinner, or bowing to the Kamiza, or whatever).
My impression has always been that most Japanese don't really care all that much whether or not you bow - it's just the custom, that's all. If you have some kind of a reason for not bowing they most likely wouldn't give it a second thought,