NavList:

David Pike, you wrote:
"I don’t remember mentioning great circles"

Oh, well, you brought up this scenario immediately after describing great circles crossing the equator in a prior post. But a rhumbline, of course, is easier so we can talk about that first.

You also said you didn't mention iPads. No, I tossed that into your scenario (strictly for laughs) to remind everyone that the days of depending on the gear in the cockpit are as antique as a paper Mercator chart for navigation. It's an interesting phenomenon: every commercial pilot I've spoken to seems to have an iPad, and those that I have asked also say that almost everyone carries one and gets great advantage from apps and data.

You added:
"I chose Bermuda to St Helena carefully.  It means the whole flight is over the sea"

Oh yes, I understood that, and it aligns nearly with something that I have mentioned before regarding the view from the south end of the island here. Looking southeast from here, we see the open Atlantic. If you sail off in that direction and maintain a heading always parallel to your own wake, such that your course is as close to a straight line as possible (in other words travel on a great circle), you'll pass right between South America and Africa and keep right on going through the Southern Ocean into the Indian past the antipodes and won't hit land until Australia. So yes, this stretch is great for these imaginary "out of sight of land" problems. :)

For one approach to the equator-crossing rhumbline, you suggested:
"Therefore, you can navigate on your North Atlantic Mercator chart as far as the Equator, turn it upside down, renumber the longitudes, and carry on navigating to St Helena.  You’ll also need to mark in the position of St Helena and the position of any navigation aids and diversion aids you might need to use on the renumbered grid. You’ll need to renumber the longitudes to keep the first and second parts of the track a reasonable distance apart.  They’ll end up parallel to each other.  All you need to know is where you’ll cross the Equator. "

Alternatively, treat the equator as a mirror. We're looking for the straight line on the Mercator chart from Bermuda to St. Helena. So just plot your pseudo-St.H. at the correct longitude but with north latitude. Then, suddenly, this may sound familiar. It's your math-y puzzle from last year about a shortest route to the river (and maybe that was your intention!). And like that problem, it's identical to a light reflection problem. And that means it can also be solved by a "shortest string" problem or by equating angle of incidence and angle of reflection. For the latter solution, you can do it visually, and if you've played pool/billiards, you can see it easily enough: set your ball at Bermuda, bounce your shot off the equator to strike pseudo-St.H. Alternatively, if a piece of string is handy, run it from Bermuda to a (movable) pin and then up to pseudo-St.H. Slide the pin on the equator east-west until you get the shortest possible length of string at the end. You will find, of course, that this is exactly where the angle between the string and the equator on the west side is the same as the angle between string and equator on the east side. 

You noted:
"Overall, going by rhumb line adds only 24nm"

Yes, it's a relatively "easy" (mostly) north-south separation between the points which does not benefit much from worries about great circles. And certainly weather-routing would erase any geometric concerns. If you want a more trying case, consider flying from Bermuda to Îles Kerguelen. The initial heading from Bermuda should be quite similar. I bet Skinflint Air could have a monopoly on that route if they could get a license from the French (though they would surely require in-flight refueling, which might be outside their budget). It's a much longer trip, and the difference between rhumbline and great circle should be more interesting.

Frank Reed