I like the idea of the normal chat just shown over the speaker.
For the chat activation, I think enter is the best solution. But I like very much those \'pre-written sentences\' on the F1-F12. So, I was thinking, you could have, for example \'I\' to open the inventory; and, if you are typing at that moment, just CTRL+I should do the same. In other words, every game command could have an alias being the same letter plus CTRL, which would be the only way to keyborad-reach it when you are chatting.
And beware of the special keys like @ or ~, they can be easy to reach in an english keyboard, but not in averyone. for example, to type ~ in a spanish keyboard, you have to do: ALT+\'126\'.