Earlier quoted context omitted.
Presumably the mid-point is where the ship would be travelling at maximum velocity, so the course correction may not be as minimal depending on how slowly we're talking. It might be this would use more fuel/mass.
There's no such thing as maximum velocity. If constant 1g acceleration is available, there's not really a fuel concern. It will certainly use a little more fuel, but even just going to Mars would take a week, so half an hour's extra fuel usage works out to 0.3% extra fuel. For interstellar journeys, you're talking less than 0.001%. Also, the course correction can easily be part of the manoeuvre, if you're willing to…
Yeah, I could imagine some spiraling maneuver that would rotate 180° while maintaining constant acceleration. However, a flip and burn would certainly be more efficient (no cosine losses).