It's hard to see why an american, or any other race would be offended by typing "Jap." In the age of internet, everything gets shortend, using it verbally might be a different story.....
That said, only a Japanese person could decide if that word is offensive to them. If it is, it's hard to argue that it shouldn't be offensive to them. You can say you didn't mean it in an offensive way, and they might tell they understand but do not appreciate being called in that manner. Then again, maybe not.
Guess it is a fine line, and with only brief words, things and meanings that are not intended can be interpreted different ways. Personally, I know there may be an issue with the word "Jap," so therefore I just type the extra 5 letters (takes less than 5 seconds) to avoid the whole situation to begin with. Same time, I would call the British "Brits," only because there doesn't seem (to my knowledge) to be much controversy around it.
Not everything in life follows what seems to be logically arguments. British to "Brit", and Japanese to "Jap" seem to not always translate in the same way. I guess it can be maddening for it's lack of logic, especially with PC nuts on the loose. When I was in my fraternity days, people would say stuff like "so you joined a frat." to which we would generally reply, "do you call your country a c**t." A bit crude I know, just thought I'd throw that in there.
In short, to non-Japanese people it may seem like a no brainer, but we are dealing with another culture. Personally, I would rather spend 3 seconds and type 5 extra letters, just to make an issue a non-issue....though as an American I would not find the word offensive.