Steve - Do you live near Carnarvon? or are you on a boat between Bermuda and The Virgin Islands.
I don't live on a boat
Goode's Homolosine projection & azimuthal equidistant projection
It is very important to use a projection where straight lines represent the shortest route, and where distances are preserved. The obvious complication is that one of the poles is involved, and very few maps will present an image which allows for an area around the pole to be seen as well as areas beyond the equator.
The best option is actually a map in the form of a 3 dimensional object.
Goode's Homolosine, for example, would rely on you passing near, or through some f the discontinuities, and fairly clearly it is not a route one would actually travel.
The Azimuthal equidistant projection allows for measurement from the centre of the map along straight lines. These maps are also known as great circle maps. An important consideration is where the centre of this projection is. Normally it must be on one of the end points of your journey, however it may also be on a point along the intended route. I would be interested in where you had the projection centred. (The south pole would have been a reasonable option)
I suspect the Carnarvon guess comes from the latter projection. It is incorrect, but it is as close as anyone who has followed my instructions has managed.
It's probably reasonable though because my measurements were done by guestimate, and also it is hard to pick a particular point on the Ronne ice shelf (fixed landmarks are somewhat of a tricky issue).
I suggest that you need to consider making your measurement in the depth of winter
I live in a capital city (and that should give it away)