I am also American, in Canada on an open work permit, because my European partner is a grad student in Canada.
An LMIA is, of course, the fastest path to an ITA. But most requests for LMIAs are rejected - I've heard as much as 80% - so you have to be careful. The LMIA is basically an assessment, after your employer advertises your job and interviews people who already have status in Canada, that a Canadian who can perform your job, or be trained to perform it, can't be found. Obviously, this is a much easier task in very technical occupations, or in geographically outlying places. With a job like mine (literary agent in Toronto) there are, of course, many Canadians who would want my job and who could be trained to do it. I was hired because of my very specific background in the arts, business, and journalism, and because I can work in Los Angeles as well as Toronto, but my employer would be hard pressed to explain why there was absolutely no Canadian who could do the same.
The first immigration lawyer I saw also recommended applying for an LMIA, but the second, more honest, one told me my chances of getting it approved were next to nothing, and I shouldn't waste my money or put my job in jeopardy. Remember that you pay the lawyer to do the work - but if you are rejected only you pay the consequences!
I applied for CEC last November, but was too late - the cap had been reached and my application was returned. In February, I posted my EE profile, with 466 points (no LMIA!), and on March 27, I got an ITA in the FSW class, which is fine with me.
My point is that having an LMIA virtually assures an invitation to apply, but if you are in a job that will probably be turned down, get your credentials and English proficiency evaluated, post your score, and wait. You'll probably score high enough to be selected anyway.