I did just such a job DBY. My solution was to explode the profile line, and extend two approximately-half portions of it past the work sufficient to clear the tool diameter, then mill those two lines as separate MOps, moving the clamps from one side to the other between MOps. Actually, I didn't move the clamps, I installed a new set before removing the prior set, so as not to disturb the work's position.
One of the profiles was a simple 'outside L-shape' so I did that one as a simple line. The other line was more complex, so I converted to one edge of a pocket slightly larger than the tool width, so it could mill the entire profile in 'depth-first' mode without ever lifting the tool.
I arranged the two portions so they passed the edges of the work at corners, so as not to create any visible joints in the milling paths.
Lloyd