Measuring is a good thing. However I've never been really comfortable with the direction to measure in the center, top, bottom, and average. All three should measure the same (or very, very close to the same). If they don't, there's another problem going on. It seems to me that the first priority should be to fix that problem.
As far as what to expect with a longarmer, that really depends on how bad the ruffling is (it doesn't look that bad in your picture) and how experienced the longarmer is. It could be that the finished quilt won't lie flat, or that it won't be completely square, but while it's in use that won't really matter. The problem is more likely to show up when you lay the quilt out to be photographed, or when it's hung in a show. If you block the quilt (possibly necessary every time it's washed) you may be able to force it to be flat and square.
Your quilt is very, very pretty. I would encourage you to take the borders off again, measure carefully, trim to size, and sew them on again without ruffles. I think the quilt deserves that. And you shouldn't give up on yourself by accepting something that you know can be fixed. You might try using a walking foot when sewing on the borders, definitely pin in lots of places, and (if necessary) place the fabric that needs to be eased against the feed dogs. But if you really don't want to take the borders off yet again (and I can understand your frustration) then just ask the longarmer whether she thinks the ruffles can be quilted out, and whether she has experience in doing that. And ask yourself if it's okay with you if the quilt's edges don't lie flat. (Probably no one will ever notice, unless you show them.)