i think clients have that expectation everywhere... that because FCP is cheaper than Media Composer, then the person hired to cut on it would also be cheaper. in LA it's getting much better though. i don't run into that as much any more.
and I also charge a day rate... i've charged anywhere from $350/day when i started out, to $1200/day on the rare occasion (high profile trailers). lately, i probably average $600/day plus I charge a $175/day bay rental fee as well as a $50/day media storage fee when the project requires over a TB of space. this might sound like a lot but the budgets for what i'm cutting are usually in the ballpark of $30,000 for a 10 minute piece.
In our market FCP suites are at no different of a rate than the mid level avid suites...150-200 per hour. Freelancers get the same no matter what kind of buttons they are pushing that day... national average is 50 bucks an hour. It varies with market, experience, networkability, ect.