To be honest, I don't really follow what you specifically mean by either option, but I think you might be getting way too caught up in finding the optimal approach that you're missing the point of the exercise. What we're doing here - the constructional drawing exercise of drawing this object - is solving a puzzle. The puzzle involves elements that exist in 3D space, and by solving it (by which I mean coming to some final result, not the "most correct" one as the result itself does not matter) we are forcing our brains to think through relationships that exist in 3D space, but are represented on a flat, two dimensional page.

Getting the most barrel-like form for your barrel isn't all that important in the grand scheme of things - thinking through how the different elements of the barrel exist in 3D space and ultimately coming to a decision for each mark you wish to make (and as you noted, not redrawing those decisions once they've been made) and building upon it from there on the other hand is.

So, approach the work in whatever way makes the most sense to you, while adhering to the instructions as closely as you're able, and leave the rest to be addressed in your critique.